where Δ=∏i=1nL(i,ηE/Fi)Δ=âˆi=1n Li,ηE/FiDelta=prod_(i=1)^(n)L(i,eta_(E//F)^(i))\Delta=\prod_{i=1}^{n} L\left(i, \eta_{E / F}^{i}\right)Δ=âˆi=1nL(i,ηE/Fi) and L(s,π,Ad)=∏vL(s,πv,Ad)L(s,Ï€,Ad)=âˆv Ls,Ï€v,AdL(s,pi,Ad)=prod_(v)L(s,pi_(v),Ad)L(s, \pi, A d)=\prod_{v} L\left(s, \pi_{v}, A d\right)L(s,Ï€,Ad)=âˆvL(s,Ï€v,Ad) denotes the completed adjoint LLLLL-function of πÏ€pi\piÏ€.
Note that at a formal level, that is, formally expanding LLLLL-functions as Euler products outside the range of convergence, the above formula can be rewritten in the more compact way as
where the prime symbol on the product sign indicates that it is not convergent and has to be suitably reinterpreted "in the sense of LLLLL-functions" as identity (2.3).
Thanks to the work of many authors that we are going to summarize in the next sections, it is now relatively easy to state the current status on these two conjectures:
Theorem 2.1. Both Conjectures 2.1 and 2.2 hold in full generality.
The rest of this paper is devoted to reviewing the long series of works leading to the above theorem. They all stem from a strategy originally proposed by Jacquet and Rallis [32] of comparing two relative trace formulae. Let us mention here that there has actually been other fruitful approaches to the global Gan-Gross-Prasad conjecture among which one of the most notable has been the method pioneered by Ginzburg-Jiang-Rallis [25] using automorphic descent and that has recently seen much development with the work [33] of Jiang and L. Zhang proving in full generality the implication (2) ⇒⇒=>\Rightarrow⇒ (1) of Conjecture 2.1.
2.2. The approach of Jacquet-Rallis
In [32], Jacquet and Rallis have proposed a way to attack the Gan-Gross-Prasad conjecture for unitary groups through a comparison of relative trace formulae. They only consider the case where dim(W)=dim(V)−1dimâ¡(W)=dimâ¡(V)−1dim(W)=dim(V)-1\operatorname{dim}(W)=\operatorname{dim}(V)-1dimâ¡(W)=dimâ¡(V)−1 (in which case H=U(W)H=U(W)H=U(W)H=U(W)H=U(W) and the character ξξxi\xiξ is trivial) and we assume throughout that this condition is satisfied. The global relative trace formulae considered here are powerful analytic tools introduced originally by Jacquet and that relate automorphic periods to more geometric distributions known as relative orbital integrals.
Let us be more specific in the case at hand. For f∈Cc∞(G(AF))f∈Cc∞GAFf inC_(c)^(oo)(G(A_(F)))f \in C_{c}^{\infty}\left(G\left(\mathbb{A}_{F}\right)\right)f∈Cc∞(G(AF)), a global test function, we let
Kf(x,y)=∑γ∈G(F)f(x−1γy),x,y∈G(F)∖G(AF)Kf(x,y)=∑γ∈G(F) fx−1γy,x,y∈G(F)∖GAFK_(f)(x,y)=sum_(gamma in G(F))f(x^(-1)gamma y),quad x,y in G(F)\\G(A_(F))K_{f}(x, y)=\sum_{\gamma \in G(F)} f\left(x^{-1} \gamma y\right), \quad x, y \in G(F) \backslash G\left(\mathbb{A}_{F}\right)Kf(x,y)=∑γ∈G(F)f(x−1γy),x,y∈G(F)∖G(AF)
be its automorphic kernel which describes the operator R(f)R(f)R(f)R(f)R(f) of right convolution by fffff on the space of automorphic forms. The first trace formula introduced by Jacquet and Rallis is formally obtained by expanding the (usually divergent) expression
(2.5)J(f)=∫[H]×[H]Kf(h1,h2)dh1dh2(2.5)J(f)=∫[H]×[H] Kfh1,h2dh1dh2{:(2.5)J(f)=int_([H]xx[H])K_(f)(h_(1),h_(2))dh_(1)dh_(2):}\begin{equation*}
J(f)=\int_{[H] \times[H]} K_{f}\left(h_{1}, h_{2}\right) d h_{1} d h_{2} \tag{2.5}
\end{equation*}(2.5)J(f)=∫[H]×[H]Kf(h1,h2)dh1dh2
in two different ways. More precisely, but still at a formal level, this distribution can be expanded as
where the right sum runs over an orthonormal basis for the space of cuspidal automorphic forms whereas the left sum is indexed by the so-called regular semisimple double cosets of H(F)H(F)H(F)H(F)H(F) in G(F)G(F)G(F)G(F)G(F). Here, an element δ∈Gδ∈Gdelta in G\delta \in Gδ∈G is called (relatively) regular semisimple if its stabilizer under the H×HH×HH xx HH \times HH×H-action is trivial and the corresponding orbit is (Zariski) closed. We denote by GrsGrsG_(rs)G_{r s}Grs the nonempty Zariski open subset of regular semisimple elements and for δ∈Grs(F)δ∈Grs(F)delta inG_(rs)(F)\delta \in G_{\mathrm{rs}}(F)δ∈Grs(F),
O(δ,f)=∫H(AF)×H(AF)f(h1δh2)dh1dh2O(δ,f)=∫HAF×HAF fh1δh2dh1dh2O(delta,f)=int_(H(A_(F))xx H(A_(F)))f(h_(1)deltah_(2))dh_(1)dh_(2)O(\delta, f)=\int_{H\left(\mathbb{A}_{F}\right) \times H\left(\mathbb{A}_{F}\right)} f\left(h_{1} \delta h_{2}\right) d h_{1} d h_{2}O(δ,f)=∫H(AF)×H(AF)f(h1δh2)dh1dh2
denotes the corresponding relative orbital integral of fffff at δδdelta\deltaδ. The left suspension points in (2.6) represent the remaining contributions from singular orbits whereas the right suspension points indicate the contribution of the continuous spectrum (both of which are somehow responsible for the divergence of the original expression (2.5)).
The second trace formula introduced by Jacquet and Rallis has to do with the following triple of groups:
where n=dim(W)n=dimâ¡(W)n=dim(W)n=\operatorname{dim}(W)n=dimâ¡(W), the first embedding is the diagonal one and the second embedding is the natural one. Note that G′G′G^(')G^{\prime}G′ is the group on which the base-change πEÏ€Epi_(E)\pi_{E}Ï€E "lives." For f′∈Cc∞(G′(AF))f′∈Cc∞G′AFf^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′∈Cc∞(G′(AF)), we write (again formally)
(2.7)I(f′)=∫[H1]×[H2]Kf′(h1,h2)η(h2)dh1dh2(2.7)If′=∫H1×H2 Kf′h1,h2ηh2dh1dh2{:(2.7)I(f^('))=int_([H_(1)]xx[H_(2)])K_(f^('))(h_(1),h_(2))eta(h_(2))dh_(1)dh_(2):}\begin{equation*}
I\left(f^{\prime}\right)=\int_{\left[H_{1}\right] \times\left[H_{2}\right]} K_{f^{\prime}}\left(h_{1}, h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2} \tag{2.7}
\end{equation*}(2.7)I(f′)=∫[H1]×[H2]Kf′(h1,h2)η(h2)dh1dh2
where Kf′Kf′K_(f^('))K_{f^{\prime}}Kf′ is the automorphic kernel of f′f′f^(')f^{\prime}f′ and η:[H2]→{±1}η:H2→{±1}eta:[H_(2)]rarr{+-1}\eta:\left[H_{2}\right] \rightarrow\{ \pm 1\}η:[H2]→{±1} is the automorphic character defined by η(gn,gn+1)=ηE/F(detgn)n+1ηE/F(detgn+1)nηgn,gn+1=ηE/Fdetâ¡gnn+1ηE/Fdetâ¡gn+1neta(g_(n),g_(n+1))=eta_(E//F)(det g_(n))^(n+1)eta_(E//F)(det g_(n+1))^(n)\eta\left(g_{n}, g_{n+1}\right)=\eta_{E / F}\left(\operatorname{det} g_{n}\right)^{n+1} \eta_{E / F}\left(\operatorname{det} g_{n+1}\right)^{n}η(gn,gn+1)=ηE/F(detâ¡gn)n+1ηE/F(detâ¡gn+1)n. This formal distribution can be analogously expanded as
where Grs′Grs′G_(rs)^(')G_{r s}^{\prime}Grs′ stands for the open subset of regular and semisimple elements under the H1×H2H1×H2H_(1)xxH_(2)H_{1} \times H_{2}H1×H2 action, the relative orbital integrals are given by
Oη(γ,f′)=∫H1(AF)×H2(AF)f′(h1γh2)η(h2)dh1dh2Oηγ,f′=∫H1AF×H2AF f′h1γh2ηh2dh1dh2O_(eta)(gamma,f^('))=int_(H_(1)(A_(F))xxH_(2)(A_(F)))f^(')(h_(1)gammah_(2))eta(h_(2))dh_(1)dh_(2)O_{\eta}\left(\gamma, f^{\prime}\right)=\int_{H_{1}\left(\mathbb{A}_{F}\right) \times H_{2}\left(\mathbb{A}_{F}\right)} f^{\prime}\left(h_{1} \gamma h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2}Oη(γ,f′)=∫H1(AF)×H2(AF)f′(h1γh2)η(h2)dh1dh2
and PH1,PH2,ηPH1,PH2,ηP_(H_(1)),P_(H_(2),eta)\mathscr{P}_{H_{1}}, \mathscr{P}_{H_{2}, \eta}PH1,PH2,η denote the automorphic period integrals over [H1]H1[H_(1)]\left[H_{1}\right][H1] and [H2]H2[H_(2)]\left[H_{2}\right][H2] twisted by ηηeta\etaη, respectively.
The discussion so far is, of course, oversimplifying and ignoring serious analytical and convergence issues (we will come back to this later). However, as a motivation for considering this relative trace formula on G′G′G^(')G^{\prime}G′, we have the following results on automorphic periods:
The period PH1PH1P_(H_(1))\mathscr{P}_{H_{1}}PH1 is a Rankin-Selberg period studied by Jacquet-PiatetskiiShapiro-Shalika that essentially represents the central value L(12,Π)L12,Î L((1)/(2),Pi)L\left(\frac{1}{2}, \Pi\right)L(12,Î ) on Π↪Acusp (G′)Π↪Acusp G′Pi↪A_("cusp ")(G^('))\Pi \hookrightarrow \mathscr{A}_{\text {cusp }}\left(G^{\prime}\right)Π↪Acusp (G′);
The period PH2,ηPH2,ηP_(H_(2),eta)\mathcal{P}_{H_{2}, \eta}PH2,η was studied by Rallis and Flicker who have shown that it detects exactly the cuspidal automorphic ΠÎ Pi\PiÎ 's that come by base-change from GGGGG (i.e., it is nonzero precisely on those cuspidal representations of the form πEÏ€Epi_(E)\pi_{E}Ï€E, for πÏ€pi\piÏ€ a cuspidal automorphic representation of GGGGG ).
Thus, on a very formal and sketchy sense, the Gan-Gross-Prasad conjecture implies that the spectral sides of I(f′)If′I(f^('))I\left(f^{\prime}\right)I(f′) should somehow "match" that of J(f)J(f)J(f)J(f)J(f). The idea of Jacquet and Rallis was to make precise the existence of such a comparison, from which the global GanGross-Prasad conjecture was eventually to be deduced, by equalling the geometric sides term by term. As a first step, they define a correspondence of orbits, which here takes the form of a natural embedding between regular semisimple cosets
for every field extension k/Fk/Fk//Fk / Fk/F. Using this correspondence, they then introduced a related notion of local transfer (or matching): for a place vvvvv of FFFFF, two test functions fv∈Cc∞(Gv)fv∈Cc∞Gvf_(v)inC_(c)^(oo)(G_(v))f_{v} \in C_{c}^{\infty}\left(G_{v}\right)fv∈Cc∞(Gv) and fv′∈Cc∞(Gv′)fv′∈Cc∞Gv′f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv′∈Cc∞(Gv′) are said to be transfers of each other (simply denoted by fv↔fv′fv↔fv′f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ for short) if for every δ∈H(Fv)∖Grs(Fv)/H(Fv)δ∈HFv∖GrsFv/HFvdelta in H(F_(v))\\G_(rs)(F_(v))//H(F_(v))\delta \in H\left(F_{v}\right) \backslash G_{\mathrm{rs}}\left(F_{v}\right) / H\left(F_{v}\right)δ∈H(Fv)∖Grs(Fv)/H(Fv) we have an identity
As in the usual paradigm of endoscopy, to make this notion useful and allow for a global comparison we basically need two local ingredients: first the existence of local transfer (i.e., for every fv∈Cc∞(Gv)fv∈Cc∞Gvf_(v)inC_(c)^(oo)(G_(v))f_{v} \in C_{c}^{\infty}\left(G_{v}\right)fv∈Cc∞(Gv) there exists fv′∈Cc∞(Gv′)fv′∈Cc∞Gv′f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv′∈Cc∞(Gv′) such that fv↔fv′fv↔fv′f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ and conversely, every fv′fv′f_(v)^(')f_{v}^{\prime}fv′ admits a transfer fvfvf_(v)f_{v}fv ) and then a fundamental lemma (saying, at least, that 1G(Ov)↔1G′(Ov)1GOv↔1G′Ov1_(G(O_(v)))harr1_(G^(')(O_(v)))\mathbf{1}_{G\left(\mathcal{O}_{v}\right)} \leftrightarrow \mathbf{1}_{G^{\prime}\left(\mathcal{O}_{v}\right)}1G(Ov)↔1G′(Ov) for almost all v)v{:v)\left.v\right)v).
3. COMPARISON: LOCAL TRANSFER AND FUNDAMENTAL LEMMA
A first breakthrough on the Jacquet-Rallis approach to the Gan-Gross-Prasad conjecture was made in [57] by Wei Zhang who proved the existence of the local transfer at all non-Archimedean places. His strategy for doing so roughly goes as follows:
The first step is to reduce to a statement on Lie algebras using some avatar of the exponential map (also known as Cayley map): we are then left with proving the existence of a similar transfer between the orbital integrals for the adjoint action of U(Wv)UWvU(W_(v))U\left(W_{v}\right)U(Wv) on u(Vv)=Lie(U(Vv))uVv=Lieâ¡UVvu(V_(v))=Lie(U(V_(v)))\mathfrak{u}\left(V_{v}\right)=\operatorname{Lie}\left(U\left(V_{v}\right)\right)u(Vv)=Lieâ¡(U(Vv)) and for the adjoint action of GLn(Fv)GLnFvGL_(n)(F_(v))\mathrm{GL}_{n}\left(F_{v}\right)GLn(Fv) on gn+1(Fv)gn+1Fvg_(n+1)(F_(v))\mathfrak{g}_{n+1}\left(F_{v}\right)gn+1(Fv).
Then, a crucial ingredient in Zhang's proof is to show that the transfer at the Lie algebra level essentially commutes (i.e., up to some explicit multiplicative
constants) with 3 different partial Fourier transforms F1,F2F1,F2F_(1),F_(2)\mathscr{F}_{1}, \mathscr{F}_{2}F1,F2, and F3F3F_(3)\mathscr{F}_{3}F3 that can naturally be defined on the two spaces Cc∞(u(Vv)),Cc∞(gln+1(Fv))Cc∞uVv,Cc∞gln+1FvC_(c)^(oo)(u(V_(v))),C_(c)^(oo)(gl_(n+1)(F_(v)))C_{c}^{\infty}\left(\mathfrak{u}\left(V_{v}\right)\right), C_{c}^{\infty}\left(\mathfrak{g l}_{n+1}\left(F_{v}\right)\right)Cc∞(u(Vv)),Cc∞(gln+1(Fv)). One of them, that we will denote by F1F1F_(1)\mathscr{F}_{1}F1, is the Fourier transform with respect to "the last row and column" on gln+1(Fv)gln+1Fvgl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) or u(Vv)uVvu(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv) when realizing the latter in matrix form using a basis adapted to the decomposition Vv=Wv⊕Wv⊥Vv=Wv⊕Wv⊥V_(v)=W_(v)o+W_(v)^(_|_)V_{v}=W_{v} \oplus W_{v}^{\perp}Vv=Wv⊕Wv⊥. (Recall that we are assuming that dim(Wv⊥)=1dimâ¡Wv⊥=1dim(W_(v)^(_|_))=1\operatorname{dim}\left(W_{v}^{\perp}\right)=1dimâ¡(Wv⊥)=1.) For this, Zhang develops some relative trace formulae for the aforementioned actions on gln+1(Fv)gln+1Fvgl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) and u(Vv)uVvu(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv) and combines them with a clever induction argument.
Finally, the proof of the existence of transfer on Lie algebras is obtained by combining the second step with a certain uncertainty principle due to Aizenbud [1], which allows reducing the construction of the transfer to functions that are supported away from the relative nilpotent cones (i.e., the set of elements whose orbit closure contains an element of the center of the Lie algebra), as well as a standard descent argument whose essence goes back to Harish-Chandra.
It is noteworthy to mention that this result was subsequently extended, following the same strategy, by H. Xue [53] to Archimedean places, although the final result there is slightly weaker. (More precisely, Xue was only able to show that a dense subspace of test functions admit a transfer but also observed that it is sufficient for all expected applications.)
The Jacquet-Rallis fundamental lemma for its part, was proven earlier by Yun [55] in the case of fields of positive characteristic following and adapting the geometriccohomological approach based on Hitchin fibrations that was developed by NgôNgôNgô\mathrm{Ngô}ôNgô in the context of the endoscopic fundamental lemma. This result was then transferred to fields of characteristic zero, but of sufficiently large residual characteristic, using model-theoretic techniques by Julia Gordon in the appendix of [55].
Later, in [14], I found a completely new and elementary proof of this fundamental lemma. The argument, despite that of Gordon-Yun, works directly in characteristic zero and is purely based on techniques from harmonic analysis. Thus, we have:
Theorem 3.1 (Yun-Gordon, Beuzart-Plessis). Let vvvvv be a place of FFFFF of residue characteristic not 2 that is unramified in EEEEE and assume that the Hermitian spaces Wv,Wv⊥Wv,Wv⊥W_(v),W_(v)^(_|_)W_{v}, W_{v}^{\perp}Wv,Wv⊥ both admit self-dual lattices LvWLvWL_(v)^(W)L_{v}^{W}LvW and LvW⊥LvW⊥L_(v)^(W^(_|_))L_{v}^{W^{\perp}}LvW⊥. Set Lv=LvW⊕LvW⊥Lv=LvW⊕LvW⊥L_(v)=L_(v)^(W)o+L_(v)^(W^(_|_))L_{v}=L_{v}^{W} \oplus L_{v}^{W^{\perp}}Lv=LvW⊕LvW⊥ (a self-dual lattice in VvVvV_(v)V_{v}Vv ) and Kv=StabGv(Lv×LvW)Kv=StabGvâ¡Lv×LvWK_(v)=Stab_(G_(v))(L_(v)xxL_(v)^(W))K_{v}=\operatorname{Stab}_{G_{v}}\left(L_{v} \times L_{v}^{W}\right)Kv=StabGvâ¡(Lv×LvW) for the stabilizer in Gv=U(Vv)×U(Wv)Gv=UVv×UWvG_(v)=U(V_(v))xx U(W_(v))G_{v}=U\left(V_{v}\right) \times U\left(W_{v}\right)Gv=U(Vv)×U(Wv) of the lattices LvLvL_(v)L_{v}Lv and LvWLvWL_(v)^(W)L_{v}^{W}LvW (a hyperspecial compact subgroup of GvGvG_(v)G_{v}Gv ). Then, setting Kv′=GLn+1(OEv)×Kv′=GLn+1â¡OEv×K_(v)^(')=GL_(n+1)(O_(E_(v)))xxK_{v}^{\prime}=\operatorname{GL}_{n+1}\left(\mathcal{O}_{E_{v}}\right) \timesKv′=GLn+1â¡(OEv)×GLn(OEv)GLnâ¡OEvGL_(n)(O_(E_(v)))\operatorname{GL}_{n}\left(\mathcal{O}_{E_{v}}\right)GLnâ¡(OEv), we have 1Kv↔1Kv′1Kv↔1Kv′1_(K_(v))harr1_(K_(v)^('))\mathbf{1}_{K_{v}} \leftrightarrow \mathbf{1}_{K_{v}^{\prime}}1Kv↔1Kv′.
More precisely, in [14] I proved a Lie algebra analog of the Jacquet-Rallis fundamental lemma (of which the original statement can easily be reduced; at least in residual characteristic not 2) stating that the relative orbital integrals of 1u(Lv)1uLv1_(u(L_(v)))\mathbf{1}_{\mathfrak{u}\left(L_{v}\right)}1u(Lv) match those of 1gln+1(OFv)1gln+1OFv1_(gl_(n+1))(O_(F_(v)))\mathbf{1}_{\mathfrak{g} \mathfrak{l}_{n+1}}\left(\mathcal{O}_{F_{v}}\right)1gln+1(OFv) in a suitable sense (where u(Lv)uLvu(L_(v))u\left(L_{v}\right)u(Lv) denotes the lattice in u(Vv)uVvu(V_(v))u\left(V_{v}\right)u(Vv) stabilizing LvLvL_(v)L_{v}Lv ). The argument is based on a hidden SL(2) symmetry involving a Weil representation. More specifically, we consider the Weil representations of SL(2,Fv)SLâ¡2,FvSL(2,F_(v))\operatorname{SL}\left(2, F_{v}\right)SLâ¡(2,Fv) associated to the quadratic form qqqqq sending a
matrix of size n+1n+1n+1n+1n+1,
X=(Abcλ)X=AbcλX=([A,b],[c,lambda])X=\left(\begin{array}{ll}
A & b \\
c & \lambda
\end{array}\right)X=(Abcλ)
either in gln+1(Fv)gln+1Fvgl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) or in u(Vv)uVvu(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv), to q(X)=cbq(X)=cbq(X)=cbq(X)=c bq(X)=cb (where here, AAAAA is a square-matrix, bbbbb is a column vector, and ccccc a row vector all of size nnnnn ). Using the aforementioned result of Zhang that the transfer commutes with the partial Fourier transform F1F1F_(1)\mathscr{F}_{1}F1, it can be shown that these representations descend to spaces of relative orbital integrals on Cc∞(u(Vv))Cc∞uVvC_(c)^(oo)(u(V_(v)))C_{c}^{\infty}\left(u\left(V_{v}\right)\right)Cc∞(u(Vv)) and Cc∞(gln+1(Fv))Cc∞gln+1FvC_(c)^(oo)(gl_(n+1)(F_(v)))C_{c}^{\infty}\left(\mathfrak{g l}_{n+1}\left(F_{v}\right)\right)Cc∞(gln+1(Fv)) and coincide on their intersections (identifying the spaces of regular semisimple orbits through a correspondence similar to (2.9)). Consider then the difference
where u(Vv)rs uVvrs u(V_(v))_("rs ")\mathfrak{u}\left(V_{v}\right)_{\text {rs }}u(Vv)rs denotes the Lie algebra analog of the relative regular semisimple locus, YYYYY is the image of XXXXX by a correspondence of orbits u(Vv)rs/U(Wv)↪gln+1(Fv)rs/GLn(Fv)uVvrs/UWv↪gln+1Fvrs/GLnFvu(V_(v))_(rs)//U(W_(v))↪gl_(n+1)(F_(v))_(rs)//GL_(n)(F_(v))\mathfrak{u}\left(V_{v}\right)_{\mathrm{rs}} / U\left(W_{v}\right) \hookrightarrow \mathfrak{g l}_{n+1}\left(F_{v}\right)_{\mathrm{rs}} / \mathrm{GL}_{n}\left(F_{v}\right)u(Vv)rs/U(Wv)↪gln+1(Fv)rs/GLn(Fv) similar to (2.9) and ωv(Y)ωv(Y)omega_(v)(Y)\omega_{v}(Y)ωv(Y) is the Lie algebra counterpart of the transfer factor. The fundamental lemma then states that ΦΦPhi\PhiΦ is identically zero. The proof proceeds roughly in three steps:
Secondly, we remark that ΦΦPhi\PhiΦ is also fixed by w=(0−110)w=0−110w=([0,-1],[1,0])w=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)w=(0−110). This comes from the fact that the action of wwwww descends from the partial Fourier transform F1F1F_(1)\mathscr{F}_{1}F1 which leaves (for a suitable normalization) the functions 1u(LvV),1gn+1(OFv)1uLvV,1gn+1OFv1_(u(L_(v)^(V))),1_(g_(n+1)(O_(F_(v))))\mathbf{1}_{\mathfrak{u}\left(L_{v}^{V}\right)}, \mathbf{1}_{\mathfrak{g}_{n+1}\left(\mathcal{O}_{F_{v}}\right)}1u(LvV),1gn+1(OFv) invariant.
Finally, as SL2(Fv)SL2FvSL_(2)(F_(v))\mathrm{SL}_{2}\left(F_{v}\right)SL2(Fv) is generated by (1pFv−101)1pFv−101([1p_(F_(v))^(-1)],[0],[1])\left(\begin{array}{c}1 \mathcal{p}_{F_{v}}^{-1} \\ 0 \\ 1\end{array}\right)(1pFv−101) and wwwww, we infer that ΦΦPhi\PhiΦ is fixed by SL2(Fv)SL2FvSL_(2)(F_(v))\mathrm{SL}_{2}\left(F_{v}\right)SL2(Fv) from which it is relatively straightforward to deduce Φ=0Φ=0Phi=0\Phi=0Φ=0.
It is also worth mentioning that in a very interesting work, Jingwei Xiao [51] has shown that the Jacquet-Rallis fundamental lemma implies the (usual) endocospic fundamental lemma for unitary groups. Thus, combining his argument with the proof outlined above yields a completely elementary proof of the Langlands-Shelstad fundamental lemma for unitary groups!
The two previous results on smooth transfer and the fundamental lemma are already enough to imply the Gan-Gross-Prasad Conjecture 2.1 under some local restrictions on the cuspidal representation πÏ€pi\piÏ€ (originating from the use of simple versions of the Jacquet-Rallis trace formulae, allowing to bypass all convergence issues) as was done by W. Zhang in [57]. However, to derive the refinement of Conjecture 2.2 following the same strategy, we need an extra local ingredient relating the local periods of Ichino-Ikeda to similar local distributions associated to the Rankin-Selberg and Flicker-Rallis periods. More precisely, by the work of Jacquet-Piatetskii-Shapiro-Shalika, on the one hand, and Flicker-Rallis, on the other hand,
it is known that the two automorphic periods PH1PH1P_(H_(1))\mathscr{P}_{H_{1}}PH1 and PH2,ηPH2,ηP_(H_(2),eta)\mathscr{P}_{H_{2}, \eta}PH2,η admit factorizations of the form
for φφvarphi\varphiφ a factorizable vector in a given cuspidal automorphic representation Π=Πn+1⊗ΠnÎ =Î n+1⊗ΠnPi=Pi_(n+1)oxPi_(n)\Pi=\Pi_{n+1} \otimes \Pi_{n}Î =Î n+1⊗Πn of G′(AF)G′AFG^(')(A_(F))G^{\prime}\left(\mathbb{A}_{F}\right)G′(AF), where
Wφ(g)=∫[N′]φ(ug)ψ′(u)−1du=∏vWφ,v(gv)Wφ(g)=∫N′ φ(ug)ψ′(u)−1du=âˆv Wφ,vgvW_(varphi)(g)=int_([N^(')])varphi(ug)psi^(')(u)^(-1)du=prod_(v)W_(varphi,v)(g_(v))W_{\varphi}(g)=\int_{\left[N^{\prime}\right]} \varphi(u g) \psi^{\prime}(u)^{-1} d u=\prod_{v} W_{\varphi, v}\left(g_{v}\right)Wφ(g)=∫[N′]φ(ug)ψ′(u)−1du=âˆvWφ,v(gv)
denotes a factorization of the Whittaker function of φφvarphi\varphiφ (here N′N′N^(')N^{\prime}N′ stands for the standard maximal unipotent subgroup of G′G′G^(')G^{\prime}G′ and ψ′ψ′psi^(')\psi^{\prime}ψ′ is a nondegenerate character of [N′]),PH1,v,PH2,η,vN′,PH1,v,PH2,η,v{:[N^(')]),P_(H_(1),v),P_(H_(2),eta,v)\left.\left[N^{\prime}\right]\right), \mathscr{P}_{H_{1}, v}, \mathscr{P}_{H_{2}, \eta, v}[N′]),PH1,v,PH2,η,v are explicit linear forms on the local Whittaker model W(Πv,ψv′)WÎ v,ψv′W(Pi_(v),psi_(v)^('))\mathcal{W}\left(\Pi_{v}, \psi_{v}^{\prime}\right)W(Î v,ψv′) of ΠvÎ vPi_(v)\Pi_{v}Î v and the products in (3.1), (3.2) are to be regularized and understood "in the sense of LLLLL-functions" in a way similar to (2.4).
Based on the factorizations (3.1) and (3.2), the contribution of ΠÎ Pi\PiÎ to the spectral expansion (2.8) can be shown to itself admit a factorization roughly as the product of local distributions (called relative characters) IΠvIÎ vI_(Pi_(v))I_{\Pi_{v}}IÎ v defined by
where the sum runs over a suitable orthonormal basis of the Whittaker model. On the other hand, from the Ichino-Ikeda Conjecture 2.2, we expect the contribution of π↪Acusp (G)π↪Acusp (G)pi↪A_("cusp ")(G)\pi \hookrightarrow \mathcal{A}_{\text {cusp }}(G)π↪Acusp (G) to the spectral expansion of (2.6) to essentially factorize into the product of the local relative characters (where again the sum is taken over an orthonormal basis)
In [56], W. Zhang has conjectured that the local Jacquet-Rallis transfer fv↔fv′fv↔fv′f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ also satisfies certain precise spectral relations involving the above relative characters. This is exactly the extra local ingredient needed to finish the proof of the Ichino-Ikeda conjecture based on a comparison of the Jacquet-Rallis trace formulae. This conjecture was shown in [56] to hold for unramified and supercuspidal representations, and the method was further extended and amplified in [13], allowing to prove the conjecture for all (tempered) representations at non-Archimedean places. Later, in [15] I gave a better proof of this conjecture which also has the advantage of working uniformly at all places (including Archimedean ones). To state the result, we introduce some terminology/notation: for a place vvvvv of FFFFF and a smooth irreducible representation πvÏ€vpi_(v)\pi_{v}Ï€v of GvGvG_(v)G_{v}Gv, we denote by πE,vÏ€E,vpi_(E,v)\pi_{E, v}Ï€E,v the local base-change of πvÏ€vpi_(v)\pi_{v}Ï€v, that is, the smooth irreducible representation of Gv′Gv′G_(v)^(')G_{v}^{\prime}Gv′ whose LLLLL-parameter is given by composing that of πvÏ€vpi_(v)\pi_{v}Ï€v with the natural embedding of LLLLL-groups LGv→LGv′LGv→LGv′^(L)G_(v)rarr^(L)G_(v)^('){ }^{L} G_{v} \rightarrow{ }^{L} G_{v}^{\prime}LGv→LGv′, and, moreover, we say that πvÏ€vpi_(v)\pi_{v}Ï€v is HvHvH_(v)H_{v}Hv-distinguished if HomHv(πv,C)≠0HomHvâ¡Ï€v,C≠0Hom_(H_(v))(pi_(v),C)!=0\operatorname{Hom}_{H_{v}}\left(\pi_{v}, \mathbb{C}\right) \neq 0HomHvâ¡(Ï€v,C)≠0, that is, with the notation of Section 1.1, if the multiplicity m(πv)mÏ€vm(pi_(v))m\left(\pi_{v}\right)m(Ï€v) equals 1 .
Theorem 3.2. There exist explicit local constants (κv)vκvv(kappa_(v))_(v)\left(\kappa_{v}\right)_{v}(κv)v indexed by the set of all places of FFFFF and satisfying the product formula ∏vκv=1âˆv κv=1prod_(v)kappa_(v)=1\prod_{v} \kappa_{v}=1âˆvκv=1 such that the following property is verified: for every place vvvvv, every tempered representation πvÏ€vpi_(v)\pi_{v}Ï€v of GvGvG_(v)G_{v}Gv which is HvHvH_(v)H_{v}Hv-distinguished and every pair (fv,fv′)∈Cc∞(Gv)×Cc∞(Gv′)fv,fv′∈Cc∞Gv×Cc∞Gv′(f_(v),f_(v)^('))inC_(c)^(oo)(G_(v))xxC_(c)^(oo)(G_(v)^('))\left(f_{v}, f_{v}^{\prime}\right) \in C_{c}^{\infty}\left(G_{v}\right) \times C_{c}^{\infty}\left(G_{v}^{\prime}\right)(fv,fv′)∈Cc∞(Gv)×Cc∞(Gv′) of matching functions (that is, fv↔fv′fv↔fv′f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ ), we have
Moreover, the above identities characterize the Jacquet-Rallis transfer, that is, if two functions fv∈Cc∞(Gv),fv′∈Cc∞(Gv′)fv∈Cc∞Gv,fv′∈Cc∞Gv′f_(v)inC_(c)^(oo)(G_(v)),f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v} \in C_{c}^{\infty}\left(G_{v}\right), f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv∈Cc∞(Gv),fv′∈Cc∞(Gv′) satisfy (3.3) for every tempered irreducible representation πvÏ€vpi_(v)\pi_{v}Ï€v of GvGvG_(v)G_{v}Gv that is HvHvH_(v)H_{v}Hv-distinguished, then these functions are transfers of each other.
The proof given in [15] of the above theorem is mainly based on another ingredient of independent interest which is an explicit Plancherel decomposition for the space Gv′/H2,vGv′/H2,vG_(v)^(')//H_(2,v)G_{v}^{\prime} / H_{2, v}Gv′/H2,v or rather, decomposing this quotient as a product in a natural way, for the symmetric variety GLn(Ev)/GLn(Fv)GLnEv/GLnFvGL_(n)(E_(v))//GL_(n)(F_(v))\mathrm{GL}_{n}\left(E_{v}\right) / \mathrm{GL}_{n}\left(F_{v}\right)GLn(Ev)/GLn(Fv). This spectral decomposition is roughly obtained by applying the Plancherel formula for the group GLn(Ev)GLnEvGL_(n)(E_(v))\mathrm{GL}_{n}\left(E_{v}\right)GLn(Ev) to a family of zeta integrals, depending on a complex parameter sssss, introduced by Flicker and Rallis [22] and that represents local Asai LLLLL-factors and taking the residue at s=1s=1s=1s=1s=1 of the resulting expression. We will not describe the exact process here, but just mention that this settles in the case at hand a general conjecture of Sakellaridis-Venkatesh [41] on the spectral decomposition of spherical varieties. This Plancherel formula is then used to write the explicit spectral expansion for a local analog of the Jacquet-Rallis trace formula (2.8) which is then compared with a local counterpart of the trace formula (2.6) yielding as a consequence Theorem 3.2 above. Moreover, as another byproduct of this local comparison, we also get a formula conjectured by Hiraga-Ichino-Ikeda for the formal degree of discrete series [29] in the case of unitary groups.
4. GLOBAL ANALYSIS OF JACQUET-RALLIS TRACE FORMULAE
With all the local ingredients explained in the previous section in place, the only remaining tasks to finish the program initiated by Jacquet and Rallis to prove the GanGross-Prasad and Ichino-Ikeda conjectures are global. More specifically, although simple versions of the Jacquet-Rallis trace formulae have been successfully used to establish these conjectures under some local restrictions [13,57], in order to detect all the relevant cuspidal representations of unitary groups, we need more refined versions of the geometric and spectral expansions of (2.6) and (2.8).
As a first important step in that direction, Zydor [58,59] has completely regularized the singular contributions to the geometric sides. We can summarize his main results as follows: for all test functions f∈Cc∞(G(AF))f∈Cc∞GAFf inC_(c)^(oo)(G(A_(F)))f \in C_{c}^{\infty}\left(G\left(\mathbb{A}_{F}\right)\right)f∈Cc∞(G(AF)) and f′∈Cc∞(G′(AF))f′∈Cc∞G′AFf^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′∈Cc∞(G′(AF)), there exist "canonical" regularization of the (usually divergent) integrals (2.5) and (2.7), that we will still denote by J(f)J(f)J(f)J(f)J(f) and I(f′)If′I(f^('))I\left(f^{\prime}\right)I(f′), as well as decompositions
(4.1)J(f)=∑δ∈(H∖G//H)(F)O(δ,f) and I(f′)=∑γ∈(H1∖G′//H2)(F)Oη(γ,f′)(4.1)J(f)=∑δ∈(H∖G//H)(F) O(δ,f) and If′=∑γ∈H1∖G′//H2(F) Oηγ,f′{:(4.1)J(f)=sum_(delta in(H\\G////H)(F))O(delta","f)quad" and "quad I(f^('))=sum_(gamma in(H_(1)\\G^(')////H_(2))(F))O_(eta)(gamma,f^(')):}\begin{equation*}
J(f)=\sum_{\delta \in(H \backslash G / / H)(F)} O(\delta, f) \quad \text { and } \quad I\left(f^{\prime}\right)=\sum_{\gamma \in\left(H_{1} \backslash G^{\prime} / / H_{2}\right)(F)} O_{\eta}\left(\gamma, f^{\prime}\right) \tag{4.1}
\end{equation*}(4.1)J(f)=∑δ∈(H∖G//H)(F)O(δ,f) and I(f′)=∑γ∈(H1∖G′//H2)(F)Oη(γ,f′)
where H∖G//HH∖G//HH\\G////HH \backslash G / / HH∖G//H and H1∖G′//H2H1∖G′//H2H_(1)\\G^(')////H_(2)H_{1} \backslash G^{\prime} / / H_{2}H1∖G′//H2 stand for the corresponding categorical quotients and O(δ,⋅),Oη(γ,⋅)O(δ,â‹…),Oη(γ,â‹…)O(delta,*),O_(eta)(gamma,*)O(\delta, \cdot), O_{\eta}(\gamma, \cdot)O(δ,â‹…),Oη(γ,â‹…) are distributions supported on the union of the adelic double cosets with images δδdelta\deltaδ and γγgamma\gammaγ in (H∖G//H)(AF)(H∖G//H)AF(H\\G////H)(A_(F))(H \backslash G / / H)\left(\mathbb{A}_{F}\right)(H∖G//H)(AF) and (H1∖G′//H2)(AF)H1∖G′//H2AF(H_(1)\\G^(')////H_(2))(A_(F))\left(H_{1} \backslash G^{\prime} / / H_{2}\right)\left(\mathbb{A}_{F}\right)(H1∖G′//H2)(AF), respectively, which coincide with the previously defined relative orbital integrals when δδdelta\deltaδ and γγgamma\gammaγ are regular semisimple.
Zydor obtains these regularized orbital integrals by adapting a truncation procedure developed by Arthur in the context of the usual trace formula to the relative setting at hand. It should be emphasized that contrary to what happens with Arthur's trace formula, the resulting distributions are directly invariant (in a relative sense, that is, here under the natural action of H×HH×HH xx HH \times HH×H or H1×H2H1×H2H_(1)xxH_(2)H_{1} \times H_{2}H1×H2 ) and do not depend on any auxiliary choice (such as that of a maximal compact subgroup). It is in this sense that the regularizations of Zydor are really "canonical." It should be mentioned that another, different, approach to such regularization was proposed by Sakellaridis [40] in the context of general relative trace formulae. It is based on analyzing the exponents at infinity of generalized theta series together with a natural procedure to regularize integrals of multiplicative functions when the corresponding character is nontrivial.
Before we even consider the analogous, more subtle, regularization problem on the spectral side, there appears the natural question of how to compare the singular contributions to the refined geometric expansions of (4.1). This issue was completely resolved in a very long paper [20] by Chaudouard and Zydor. To state their main result, it is convenient to again consider the relevant pure inner forms of GGGGG (as defined in Section 1.1): for every Hermitian space W′W′W^(')W^{\prime}W′ of the same dimension as WWWWW, we have a relevant pure inner form GW′=GW′=G^(W^('))=G^{W^{\prime}}=GW′=U(V′)×U(W′)UV′×UW′U(V^('))xx U(W^('))U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)U(V′)×U(W′) with its diagonal subgroup HW′=U(W′)HW′=UW′H^(W^('))=U(W^('))H^{W^{\prime}}=U\left(W^{\prime}\right)HW′=U(W′) where V′=W′⊕⊥W⊥V′=W′⊕⊥W⊥V^(')=W^(')o+^(_|_)W^(_|_)V^{\prime}=W^{\prime} \oplus^{\perp} W^{\perp}V′=W′⊕⊥W⊥. Moreover, the correspondence of orbits (2.9) extends to an isomorphism between categorical quotients,
(4.2)H∖G//H≃H1∖G′//H2(4.2)H∖G//H≃H1∖G′//H2{:(4.2)H\\G////H≃H_(1)\\G^(')////H_(2):}\begin{equation*}
H \backslash G / / H \simeq H_{1} \backslash G^{\prime} / / H_{2} \tag{4.2}
\end{equation*}(4.2)H∖G//H≃H1∖G′//H2
and for every W′W′W^(')W^{\prime}W′ as before, HW′∖GW′//HW′HW′∖GW′//HW′H^(W^('))\\G^(W^('))////H^(W^('))H^{W^{\prime}} \backslash G^{W^{\prime}} / / H^{W^{\prime}}HW′∖GW′//HW′ can naturally be identified with H∖G//HH∖G//HH\\G////HH \backslash G / / HH∖G//H. With these preliminaries, the main result of Chaudouard and Zydor can now be stated as follows:
Theorem 4.1 (Chaudouard-Zydor). Assume that fW′=∏vfvW′∈Cc∞(GW′(AF))fW′=âˆv fvW′∈Cc∞GW′AFf^(W^('))=prod_(v)f_(v)^(W^('))inC_(c)^(oo)(G^(W^('))(A_(F)))f^{W^{\prime}}=\prod_{v} f_{v}^{W^{\prime}} \in C_{c}^{\infty}\left(G^{W^{\prime}}\left(\mathbb{A}_{F}\right)\right)fW′=âˆvfvW′∈Cc∞(GW′(AF)), where W′W′W^(')W^{\prime}W′ runs over all isomorphism classes of Hermitian spaces of dimension nnnnn, and f′=∏vfv′∈Cc∞(G′(AF))f′=âˆv fv′∈Cc∞G′AFf^(')=prod_(v)f_(v)^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=âˆvfv′∈Cc∞(G′(AF)) are factorizable test functions such that for every place vvvvv, and each W′,fvW′W′,fvW′W^('),f_(v)^(W^('))W^{\prime}, f_{v}^{W^{\prime}}W′,fvW′ and fv′fv′f_(v)^(')f_{v}^{\prime}fv′ are Jacquet-Rallis transfers of each other (that is, fvW′↔fv′fvW′↔fv′f_(v)^(W^('))harrf_(v)^(')f_{v}^{W^{\prime}} \leftrightarrow f_{v}^{\prime}fvW′↔fv′ ). Then, for every δ∈(H∖G//H)(F)δ∈(H∖G//H)(F)delta in(H\\G////H)(F)\delta \in(H \backslash G / / H)(F)δ∈(H∖G//H)(F) with image γ∈(H1∖G′//H2)(F)γ∈H1∖G′//H2(F)gamma in(H_(1)\\G^(')////H_(2))(F)\gamma \in\left(H_{1} \backslash G^{\prime} / / H_{2}\right)(F)γ∈(H1∖G′//H2)(F) by (4.2), we have
It should be noted that when δδdelta\deltaδ, hence also γγgamma\gammaγ, is regular semisimple, the left-hand sum in (4.3) only contains one nonidentically vanishing term but that in general more than one relevant pure inner forms can contribute. Also, the above result extends to nonfactorizable test functions, provided the wording is changed suitably.
The next natural step would be to develop regularized spectral expansions similar to (4.1). As a first result in that direction, Zydor has shown decompositions of the form
(4.4)J(f)=∑χ∈X(G)Jχ(f) and I(f′)=∑χ′∈X(G′)Iχ′(f′)(4.4)J(f)=∑χ∈X(G) Jχ(f) and If′=∑χ′∈XG′ Iχ′f′{:(4.4)J(f)=sum_(chi inX(G))J_(chi)(f)quad" and "quad I(f^('))=sum_(chi^(')in X(G^(')))I_(chi^('))(f^(')):}\begin{equation*}
J(f)=\sum_{\chi \in \mathcal{X}(G)} J_{\chi}(f) \quad \text { and } \quad I\left(f^{\prime}\right)=\sum_{\chi^{\prime} \in X\left(G^{\prime}\right)} I_{\chi^{\prime}}\left(f^{\prime}\right) \tag{4.4}
\end{equation*}(4.4)J(f)=∑χ∈X(G)Jχ(f) and I(f′)=∑χ′∈X(G′)Iχ′(f′)
where X(G)X(G)X(G)\mathcal{X}(G)X(G) and X(G′)XG′X(G^('))\mathcal{X}\left(G^{\prime}\right)X(G′) stand for the set of cuspidal data of the groups GGGGG and G′G′G^(')G^{\prime}G′ respectively, that is the sets of pairs (M,σ)(M,σ)(M,sigma)(M, \sigma)(M,σ) where MMMMM is a Levi subgroup (of GGGGG or G′G′G^(')G^{\prime}G′ ) and σσsigma\sigmaσ is a cuspidal automorphic representation of M(AF)MAFM(A_(F))M\left(\mathbb{A}_{F}\right)M(AF) taken up to conjugacy (by G(F)G(F)G(F)G(F)G(F) or G′(F)G′(F)G^(')(F)G^{\prime}(F)G′(F) ). According to Langlands theory of pseudo-Eisenstein series, these sets index natural equivariant Hilbertian decompositions:
The automorphic kernels Kf,Kf′Kf,Kf′K_(f),K_(f^('))K_{f}, K_{f^{\prime}}Kf,Kf′ decompose accordingly into series Kf=∑χKf,χKf=∑χ Kf,χK_(f)=sum_(chi)K_(f,chi)K_{f}=\sum_{\chi} K_{f, \chi}Kf=∑χKf,χ, Kf′=∑χ′Kf′,χ′Kf′=∑χ′ Kf′,χ′K_(f^('))=sum_(chi^('))K_(f^('),chi^('))K_{f^{\prime}}=\sum_{\chi^{\prime}} K_{f^{\prime}, \chi^{\prime}}Kf′=∑χ′Kf′,χ′ where Kf,χKf,χK_(f,chi)K_{f, \chi}Kf,χ and Kf′,χKf′,χK_(f^('),chi)K_{f^{\prime}, \chi}Kf′,χ are kernel functions representing the restrictions Rχ(f)Rχ(f)R_(chi)(f)R_{\chi}(f)Rχ(f) and Rχ′(f′)Rχ′f′R_(chi^('))(f^('))R_{\chi^{\prime}}\left(f^{\prime}\right)Rχ′(f′) of the right convolution operators R(f)R(f)R(f)R(f)R(f) and R(f′)Rf′R(f^('))R\left(f^{\prime}\right)R(f′) to Lχ2([G])Lχ2([G])L_(chi)^(2)([G])L_{\chi}^{2}([G])Lχ2([G]) and Lχ′2([G′])Lχ′2G′L_(chi^('))^(2)([G^(')])L_{\chi^{\prime}}^{2}\left(\left[G^{\prime}\right]\right)Lχ′2([G′]), respectively. The distributions f↦Jχ(f)f↦Jχ(f)f|->J_(chi)(f)f \mapsto J_{\chi}(f)f↦Jχ(f) and f′↦Iχ′(f′)f′↦Iχ′f′f^(')|->I_(chi^('))(f^('))f^{\prime} \mapsto I_{\chi^{\prime}}\left(f^{\prime}\right)f′↦Iχ′(f′) are then roughly defined by applying the same regularization procedure that Zydor used for the expressions J(f)J(f)J(f)J(f)J(f) and I(f′)If′I(f^('))I\left(f^{\prime}\right)I(f′) up to replacing the integrands by Kf,χKf,χK_(f,chi)K_{f, \chi}Kf,χ and Kf′,χ′Kf′,χ′K_(f^('),chi^('))K_{f^{\prime}, \chi^{\prime}}Kf′,χ′ respectively, that is, in symbolic terms:
(4.5)Jχ(f)=∫[H]×[H]regKf,χ(h1,h2)dh1dh2Iχ′(f′)=∫[H1]×[H2]regKf′,χ′(h1,h2)η(h2)dh1dh2(4.5)Jχ(f)=∫[H]×[H]reg Kf,χh1,h2dh1dh2Iχ′f′=∫H1×H2reg Kf′,χ′h1,h2ηh2dh1dh2{:[(4.5)J_(chi)(f)=int_([H]xx[H])^(reg)K_(f,chi)(h_(1),h_(2))dh_(1)dh_(2)],[I_(chi^('))(f^('))=int_([H_(1)]xx[H_(2)])^(reg)K_(f^('),chi^('))(h_(1),h_(2))eta(h_(2))dh_(1)dh_(2)]:}\begin{align*}
J_{\chi}(f) & =\int_{[H] \times[H]}^{\mathrm{reg}} K_{f, \chi}\left(h_{1}, h_{2}\right) d h_{1} d h_{2} \tag{4.5}\\
I_{\chi^{\prime}}\left(f^{\prime}\right) & =\int_{\left[H_{1}\right] \times\left[H_{2}\right]}^{\mathrm{reg}} K_{f^{\prime}, \chi^{\prime}}\left(h_{1}, h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2}
\end{align*}(4.5)Jχ(f)=∫[H]×[H]regKf,χ(h1,h2)dh1dh2Iχ′(f′)=∫[H1]×[H2]regKf′,χ′(h1,h2)η(h2)dh1dh2
However, the expansions (4.4) are of little use as they stand and need to be suitably refined to allow for a meaningful comparison of the trace formulae. In Arthur's terminology, (4.4) are coarse spectral expansions and we need refined spectral expansions for each of the terms Jχ(f)Jχ(f)J_(chi)(f)J_{\chi}(f)Jχ(f) or Iχ′(f′)Iχ′f′I_(chi^('))(f^('))I_{\chi^{\prime}}\left(f^{\prime}\right)Iχ′(f′).
This problem has so far proved to be a very difficult for general cuspidal data χχchi\chiχ and χ′χ′chi^(')\chi^{\prime}χ′. However, a recent result of mine in collaboration with Y. Liu, W. Zhang, and X. Zhu [17] allows isolating in the coarse spectral expansions (4.4) the only terms that are eventually of interest consequently reducing the problem to some very particular cuspidal data χ′χ′chi^(')\chi^{\prime}χ′ of G′G′G^(')G^{\prime}G′.
The result proved in [17] is very general so let us place ourself for one moment in the framework of an arbitrary connected reductive group GGGGG over the number field FFFFF. Let ΣΣSigma\SigmaΣ be a set of non-Archimedean places of FFFFF (possibly infinite) such that for each v∈v∈v inv \inv∈ΣΣSigma\SigmaΣ, the group GvGvG_(v)G_{v}Gv is unramified and fix a hyperspecial compact subgroup Kv⊂GvKv⊂GvK_(v)subG_(v)K_{v} \subset G_{v}Kv⊂Gv with Kv=G(Ov)Kv=GOvK_(v)=G(O_(v))K_{v}=G\left(\mathcal{O}_{v}\right)Kv=G(Ov) for almost all v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ. We let XΣ(G)XΣ(G)X_(Sigma)(G)\mathcal{X}_{\Sigma}(G)XΣ(G) be the set of ΣΣSigma\SigmaΣ-unramified cuspidal data of GGGGG, that is, the cuspidal data represented by pairs (M,σ)(M,σ)(M,sigma)(M, \sigma)(M,σ) with σσsigma\sigmaσ unramified at all places of v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ (with respect to KvKvK_(v)K_{v}Kv or, rather, the hyperspecial subgroup it induces in MvMvM_(v)M_{v}Mv ). For χ∈XΣ(G)χ∈XΣ(G)chi inX_(Sigma)(G)\chi \in \mathcal{X}_{\Sigma}(G)χ∈XΣ(G), we define its ΣΣSigma\SigmaΣ-near equivalence class, henceforth denoted by NΣ(χ)NΣ(χ)N_(Sigma)(chi)\mathcal{N}_{\Sigma}(\chi)NΣ(χ), as the set of all cuspidal data χ′∈XΣ(G)χ′∈XΣ(G)chi^(')inX_(Sigma)(G)\chi^{\prime} \in \mathcal{X}_{\Sigma}(G)χ′∈XΣ(G) such that if χχchi\chiχ and χ′χ′chi^(')\chi^{\prime}χ′ are represented by pairs (M,σ)(M,σ)(M,sigma)(M, \sigma)(M,σ) and (M′,σ′)M′,σ′(M^('),sigma^('))\left(M^{\prime}, \sigma^{\prime}\right)(M′,σ′) respectively, then there exist automorphic unramified characters λλlambda\lambdaλ and λ′λ′lambda^(')\lambda^{\prime}λ′ of M(AF)MAFM(A_(F))M\left(\mathbb{A}_{F}\right)M(AF)
and M′(AF)M′AFM^(')(A_(F))M^{\prime}\left(\mathbb{A}_{F}\right)M′(AF), respectively, with the property that for every v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ the Satake parameters of the unique KvKvK_(v)K_{v}Kv-unramified subquotients in IPvGv(σv⊗λv)IPvGvσv⊗λvI_(P_(v))^(G_(v))(sigma_(v)oxlambda_(v))I_{P_{v}}^{G_{v}}\left(\sigma_{v} \otimes \lambda_{v}\right)IPvGv(σv⊗λv) and IPv′Gv(σv′⊗λv′)(IPv′Gvσv′⊗λv′I_(P_(v)^('))^(G_(v))(sigma_(v)^(')oxlambda_(v)^('))(:}I_{P_{v}^{\prime}}^{G_{v}}\left(\sigma_{v}^{\prime} \otimes \lambda_{v}^{\prime}\right)\left(\right.IPv′Gv(σv′⊗λv′)( where P,P′P,P′P,P^(')P, P^{\prime}P,P′ are arbitrary chosen parabolics with Levi components M,M′M,M′M,M^(')M, M^{\prime}M,M′ ) are isomorphic. We also fix a compact-open subgroup K=∏v∈SfKvK=âˆv∈Sf KvK=prod_(v inS_(f))K_(v)K=\prod_{v \in S_{f}} K_{v}K=âˆv∈SfKv of G(Af)GAfG(A_(f))G\left(\mathbb{A}_{f}\right)G(Af) (where SfSfS_(f)S_{f}Sf denotes the set of finite places of FFFFF and KvKvK_(v)K_{v}Kv coincides with the previous choice of hyperspecial subgroup when v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ ) and we define the Schwartz space of KKKKK-biinvariant functions on G(AF)GAFG(A_(F))G\left(\mathbb{A}_{F}\right)G(AF) as the restricted tensor product
where Cc(Kv∖Gv/Kv)CcKv∖Gv/KvC_(c)(K_(v)\\G_(v)//K_(v))C_{c}\left(K_{v} \backslash G_{v} / K_{v}\right)Cc(Kv∖Gv/Kv) denotes the space of bi- KvKvK_(v)K_{v}Kv-invariant compactly supported functions on GvGvG_(v)G_{v}Gv (that is the KvKvK_(v)K_{v}Kv-spherical Hecke algebra when v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ ), F∞F∞F_(oo)F_{\infty}F∞ is the product of the Archimedean completions of FFFFF and S(G(F∞))SGF∞S(G(F_(oo)))S\left(G\left(F_{\infty}\right)\right)S(G(F∞)) stands for the Schwartz space of the reductive Lie group G(F∞)GF∞G(F_(oo))G\left(F_{\infty}\right)G(F∞) in the sense of [19]. More precisely, S(G(F∞))SGF∞S(G(F_(oo)))\mathcal{S}\left(G\left(F_{\infty}\right)\right)S(G(F∞)) is the space of smooth functions f:G(F∞)→Cf:GF∞→Cf:G(F_(oo))rarrCf: G\left(F_{\infty}\right) \rightarrow \mathbb{C}f:G(F∞)→C such that for every polynomial differential operator on G(F∞)GF∞G(F_(oo))G\left(F_{\infty}\right)G(F∞), the derivatives DfDfDfD fDf is bounded or, equivalently, such that for every left- (or right)invariant differential operator X,XfX,XfX,XfX, X fX,Xf is decreasing faster than the inverse of any polynomial on G(F∞)GF∞G(F_(oo))G\left(F_{\infty}\right)G(F∞).
MΣ(G)=M∞(G)⨂v∈Σ′H(Gv,Kv)MΣ(G)=M∞(G)⨂v∈Σ′ HGv,KvM_(Sigma)(G)=M_(oo)(G)⨂_(v in Sigma)^(')H(G_(v),K_(v))\mathcal{M}_{\Sigma}(G)=\mathcal{M}_{\infty}(G) \bigotimes_{v \in \Sigma}^{\prime} \mathscr{H}\left(G_{v}, K_{v}\right)MΣ(G)=M∞(G)⨂v∈Σ′H(Gv,Kv)
where, for v∈Σ,H(Gv,Kv)=Cc(Kv∖Gv/Kv)v∈Σ,HGv,Kv=CcKv∖Gv/Kvv in Sigma,H(G_(v),K_(v))=C_(c)(K_(v)\\G_(v)//K_(v))v \in \Sigma, \mathscr{H}\left(G_{v}, K_{v}\right)=C_{c}\left(K_{v} \backslash G_{v} / K_{v}\right)v∈Σ,H(Gv,Kv)=Cc(Kv∖Gv/Kv) is the spherical Hecke algebra. Then, MΣ(G)MΣ(G)M_(Sigma)(G)\mathcal{M}_{\Sigma}(G)MΣ(G) acts naturally on the global Schwartz space ςK(G(AF))Ï‚KGAFÏ‚_(K)(G(A_(F)))\varsigma_{K}\left(G\left(\mathbb{A}_{F}\right)\right)Ï‚K(G(AF)), and we shall denote this action as the convolution product ∗∗***∗. One of the main result of [17] can now be stated as follows:
Theorem 4.2 (Beuzart-Plessis-Liu-Zhang-Zhu). Let χ∈XΣ(G)χ∈XΣ(G)chi inX_(Sigma)(G)\chi \in \mathcal{X}_{\Sigma}(G)χ∈XΣ(G). Then, there exists a multiplier μχ∈MΣ(G)μχ∈MΣ(G)mu_(chi)inM_(Sigma)(G)\mu_{\chi} \in \mathcal{M}_{\Sigma}(G)μχ∈MΣ(G) such that for every Schwartz function f∈SK(G(AF))f∈SKGAFf inS_(K)(G(A_(F)))f \in S_{K}\left(G\left(\mathbb{A}_{F}\right)\right)f∈SK(G(AF)) and every other cuspidal datum χ′∈XΣ(G)χ′∈XΣ(G)chi^(')inX_(Sigma)(G)\chi^{\prime} \in \mathcal{X}_{\Sigma}(G)χ′∈XΣ(G), we have
The above theorem can be roughly paraphrased by saying that the multiplier μχμχmu_(chi)\mu_{\chi}μχ "isolates" the near-equivalence class NΣ(χ)NΣ(χ)N_(Sigma)(chi)\mathcal{N}_{\Sigma}(\chi)NΣ(χ) from the other cuspidal data. A large part of the proof given in [17] consists in establishing the existence of a large subalgebra of M∞(G)M∞(G)M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) which admits an explicit spectral description, that is, through its action on irreducible Casselman-Wallach representations of G(F∞)GF∞G(F_(oo))G\left(F_{\infty}\right)G(F∞). The algebra thus constructed generalizes Arthur's multipliers [5] and, moreover, builds on previous work of Delorme [21].
Going back to the setting of the Jacquet-Rallis trace formulae, the above theorem can be applied to isolate in the expansions (4.4) the automorphic LLLLL-packet of a given cuspidal automorphic representation πÏ€pi\piÏ€ of G(AF)GAFG(A_(F))G\left(\mathbb{A}_{F}\right)G(AF), on the one hand, and the cuspidal datum χχchi\chiχ of G′G′G^(')G^{\prime}G′ "supporting" its base-change πEÏ€Epi_(E)\pi_{E}Ï€E, on the other hand. Moreover, essentially using the spectral characterization of Theorem 3.2 for the transfer, this can be done by multipliers μπ∈MΣ(G)μπ∈MΣ(G)mu_(pi)inM_(Sigma)(G)\mu_{\pi} \in \mathcal{M}_{\Sigma}(G)μπ∈MΣ(G) and μχ∈MΣ(G′)μχ∈MΣG′mu_(chi)inM_(Sigma)(G^('))\mu_{\chi} \in \mathcal{M}_{\Sigma}\left(G^{\prime}\right)μχ∈MΣ(G′) that are compatible with the Jacquet-Rallis transfer in the following sense: if f=∏vfv∈SK(G(AF))f=âˆv fv∈SKGAFf=prod_(v)f_(v)inS_(K)(G(A_(F)))f=\prod_{v} f_{v} \in S_{K}\left(G\left(\mathbb{A}_{F}\right)\right)f=âˆvfv∈SK(G(AF)) and f′=∏vfv′∈SK′(G′(AF))f′=âˆv fv′∈SK′G′AFf^(')=prod_(v)f_(v)^(')inS_(K^('))(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in S_{K^{\prime}}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=âˆvfv′∈SK′(G′(AF)) are transfers of each other then so are μπ∗fμπ∗fmu_(pi)**f\mu_{\pi} * fμπ∗f and μχ∗f′μχ∗f′mu_(chi)**f^(')\mu_{\chi} * f^{\prime}μχ∗f′ (where here we take ΣΣSigma\SigmaΣ to consist of almost all places that split in EEEEE and for K,K′K,K′K,K^(')K, K^{\prime}K,K′ arbitrary compact-open subgroups of G(Af),G′(Af)GAf,G′AfG(A_(f)),G^(')(A_(f))G\left(\mathbb{A}_{f}\right), G^{\prime}\left(\mathbb{A}_{f}\right)G(Af),G′(Af) that are hyperspecial at places in ΣΣSigma\SigmaΣ ). All in all, applying these multipliers to global test functions fffff and f′f′f^(')f^{\prime}f′ that are transfers of each other, and comparing the geometric expansions (4.1), we obtain an identity of the following shape:
where the outside left sum runs over isomorphism classes of Hermitian spaces of the same dimension as WWWWW (or, equivalently, relevant pure inner forms of GGGGG ). Besides, as a consequence of the local Gan-Gross-Prasad conjecture, when πEÏ€Epi_(E)\pi_{E}Ï€E is generic, the left-hand side contains at most one nonzero term. Thus, as a final step to establish the Gan-Gross-Prasad and IchinoIkeda conjectures, it only remains to analyze the distribution IχIχI_(chi)I_{\chi}Iχ. When the base-change πEÏ€Epi_(E)\pi_{E}Ï€E is itself cuspidal, that is, when χ={(G′,πE)}χ=G′,Ï€Echi={(G^('),pi_(E))}\chi=\left\{\left(G^{\prime}, \pi_{E}\right)\right\}χ={(G′,Ï€E)}, by the works of Jacquet-PiatetskiShapiro-Shalika and Flicker-Rallis already recalled, IχIχI_(chi)I_{\chi}Iχ essentially factors as the product of the local relative characters IπE,vIÏ€E,vI_(pi_(E,v))I_{\pi_{E, v}}IÏ€E,v and Theorem 3.2 then allows to conclude. However, in general a similar factorization of IχIχI_(chi)I_{\chi}Iχ is far from obvious and was actually established in my joint work with Chaudouard and Zydor [16]. It is exactly of the shape predicted by the Ichino-Ikeda conjecture. More precisely:
Theorem 4.3 (Beuzart-Plessis-Chaudouard-Zydor). Let πÏ€pi\piÏ€ be a cuspidal automorphic representation of G(AF)GAFG(A_(F))G\left(\mathbb{A}_{F}\right)G(AF) whose base-change πEÏ€Epi_(E)\pi_{E}Ï€E is generic. Let χχchi\chiχ be the cuspidal datum of G′G′G^(')G^{\prime}G′ such that πEÏ€Epi_(E)\pi_{E}Ï€E contributes to the spectral decomposition of Lχ2([G′])Lχ2G′L_(chi)^(2)([G^(')])L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)Lχ2([G′]). Then, for every factor izable test function f′=∏vfv′∈S(G′(AF))f′=âˆv fv′∈SG′AFf^(')=prod_(v)f_(v)^(')in S(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in S\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=âˆvfv′∈S(G′(AF)), we have
where the product has to be understood, as for (2.4), "in the sense of L-functions."
In [16], two proofs are actually given of the above theorem: one using truncations operators and the other one based on the global theory of Zeta integrals. For both methods, a crucial step is to spectrally expand the restriction of the Flicker-Rallis period (that is, the integral over [H2]H2[H_(2)]\left[H_{2}\right][H2] ) to functions φ∈Lχ2([G′])φ∈Lχ2G′varphi inL_(chi)^(2)([G^(')])\varphi \in L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)φ∈Lχ2([G′]) that are sufficiently rapidly decreasing. A consequence of this computation is that this period only depends on the πEÏ€Epi_(E)\pi_{E}Ï€E-component of φφvarphi\varphiφ and it is mainly for this reason that the contribution of χχchi\chiχ to the Jacquet-Rallis trace formula I(f′)If′I(f^('))I\left(f^{\prime}\right)I(f′) is eventually discrete (although in the case at hand, Lχ2([G′])Lχ2G′L_(chi)^(2)([G^(')])L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)Lχ2([G′]) usually has a purely continuous spectrum). For this, the truncation method is based on the work of Jacquet-Lapid-Rogawski who have defined and studied generalizations of Arthur's truncation operator to the setting of Galois periods and proved analogs of the Maass-Selberg relations in this context. On the other hand, the other method starts by expressing the Flicker-Rallis period as a residue of the integral over [H2]H2[H_(2)]\left[\mathrm{H}_{2}\right][H2] of φφvarphi\varphiφ against an Eisenstein series. Unfolding carefully this expression as in the work of Flicker-Rallis, we can rewrite it as a Zeta integral of the sort that represents Asai LLLLL-functions. The precise location of the poles of these LLLLL-functions, as well as an explicit residue computation of a family of distributions, then allows to conclude.
Finally, let me mention that in work in progress with P.-H. Chaudouard, we are able to analyze the contributions to the Jacquet-Rallis trace formula of more general cuspidal data χ∈X(G′)χ∈XG′chi inX(G^('))\chi \in \mathcal{X}\left(G^{\prime}\right)χ∈X(G′) than that appearing in Theorem [16]. The final result is similar to (4.6) except that the right-hand side has to be integrated over a certain family of automorphic representations πÏ€pi\piÏ€ of G(AF)GAFG(A_(F))G\left(\mathbb{A}_{F}\right)G(AF). More precisely, our results include some cuspidal data supporting the basechanges of automorphic representations of G=U(V)×U(W)G=U(V)×U(W)G=U(V)xx U(W)G=U(V) \times U(W)G=U(V)×U(W) that are Eisenstein in the first factor and cuspidal in the second. In this particular case, the contribution of the corresponding cuspidal datum to the trace formula J(f)J(f)J(f)J(f)J(f) is absolutely convergent and a refined spectral expansion can readily be obtained as an integral of Gan-Gross-Prasad periods between a cusp form and an Eisenstein series. These last periods are related, by some unfolding, to Bessel periods of cusp forms on smaller unitary groups. For this reason, our extension of Theorem 4.3 with Chaudouard should have similar applications to the Gan-Gross-Prasad and Ichino-Ikeda conjectures for general Bessel periods.
5. LOOKING FORWARD
As illustrated in the previous sections, various trace formula approaches to the GanGross-Prasad conjectures for unitary groups have been very successful. However, despite these favorable and definite results, these developments also raise interesting questions or have lead to fertile new research direction:
First, there is the question of whether similar techniques can be applied to prove the global Gan-Gross-Prasad conjectures for other groups. Indeed, the original conjectures in [23] also include general Bessel periods on (product of) orthogonal groups SO(n)×SO(m)(n≢m[2])SOâ¡(n)×SOâ¡(m)(n≢m[2])SO(n)xx SO(m)(n≢m[2])\operatorname{SO}(n) \times \operatorname{SO}(m)(n \not \equiv m[2])SOâ¡(n)×SOâ¡(m)(n≢m[2]), as well as so-called Fourier-Jacobi periods on unitary groups U(n)×U(m)(n≡mU(n)×U(m)(n≡mU(n)xx U(m)(n-=mU(n) \times U(m)(n \equiv mU(n)×U(m)(n≡m [2]) or symplectic/metaplectic groups Mp(n)×Sp(m)Mpâ¡(n)×Spâ¡(m)Mp(n)xx Sp(m)\operatorname{Mp}(n) \times \operatorname{Sp}(m)Mpâ¡(n)×Spâ¡(m). In the case of U(n)×U(n)U(n)×U(n)U(n)xx U(n)U(n) \times U(n)U(n)×U(n), a relative trace formula approach
has been proposed by Y. Liu and further developed by H. Xue [52]. However, the situation is not as complete as for the Jacquet-Rallis trace formulae in the case of U(n+1)×U(n)U(n+1)×U(n)U(n+1)xx U(n)U(n+1) \times U(n)U(n+1)×U(n). It would be interesting to see if the latest developments, in particular those from my joint work with Chaudouard and Zydor [16], can be adapted to this setting. This could possibly lead to a proof of the GanGross-Prasad conjecture for general Fourier-Jacobi periods on unitary groups. The situation for orthogonal and symplectic/metaplectic groups is much less satisfactory and there is no clear approach through a comparison of relative trace formulae, yet. This is due in particular to the fact that, instead of the FlickerRallis periods, in these cases we are naturally lead to consider period integrals originally studied by Bump-Ginzburg that detect cuspidal automorphic representations of GL(n)GL(n)GL(n)\mathrm{GL}(n)GL(n) of orthogonal type. These period integrals involve the product of two exceptional theta series on a double cover of GL(n)GLâ¡(n)GL(n)\operatorname{GL}(n)GLâ¡(n) and do not have any obvious geometric realizations (except when n=2n=2n=2n=2n=2 ). This makes the task of writing a geometric expansion for the corresponding trace formulae quite unclear. It would certainly be interesting to see if the recent Hamiltonian duality picture of Ben Zvi-Sakellaridis-Venkatesh can shed some light on this matter (in particular, by associating a Hamiltonian space to the Bump-Ginzburg periods).
In the local setting, the new trace formulae first discovered by Waldspurger [47] and further developed in [12] seem to be of quite broad applicability to all kind of distinction problems. Actually, similar trace formulae have already been established in other contexts [11,18,50][11,18,50][11,18,50][11,18,50][11,18,50] with new applications in the spirit of "relative Langlands functorialities" each time. However, all these developments have been made on a case-by-case basis so far and it would be very interesting and instructive to elaborate a general theory. In particular, in view of the proposal by Sakellaridis-Venkatesh [41] of a general framework for the relative Langlands program, we could hope to establish general local relative trace formulae for the L2L2L^(2)L^{2}L2 spaces of spherical varieties XXXXX and relate those to the dual group construction of Sakellaridis-Venkatesh.
Finally, in a slightly different direction the general isolation Theorem 4.2 clearly has the potential to be applied in other context, e.g., it would be interesting to see if it can be used as a technical device to simplify some other known comparison of trace formulae. Another intriguing question is to look for a precise spectral description of the (abstract) multiplier algebra M∞(G)M∞(G)M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) and in [17], we actually argue that M∞(G)M∞(G)M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) should be seen as the natural Archimedean analog of the Bernstein center for ppppp-adic groups.
ACKNOWLEDGMENTS
First and foremost, I thank my former PhD advisor Jean-Loup Waldspurger for introducing me to this fertile circle of ideas and for always being so generous in sharing his insights on these problems.
I am also grateful to my colleagues and coauthors Pierre-Henri Chaudouard, Patrick Delorme, Wee-Teck Gan, Yifeng Liu, Dipendra Prasad, Yiannis Sakellaridis, Chen Wan, Wei Zhang, Xinwen Zhu, and Michal Zydor for many inspiring discussions and exchanges over the years.
FUNDING
The author's work was partially supported by the Excellence Initiative of Aix-Marseille University (A*MIDEX), a French "Investissements d'Avenir" program.
REFERENCES
[1] A. Aizenbud, A partial analog of the integrability theorem for distributions on p-adic spaces and applications. Israel J. Math. 193 (2013), no. 1, 233-262.
[2] A. Aizenbud, D. Gourevitch, S. Rallis, and G. Schiffmann, Multiplicity one theorems. Ann. of Math. (2) 172 (2010), no. 2, 1407-1434.
[3] J. Arthur, A trace formula for reductive groups. I. Terms associated to classes in G(Q)G(Q)G(Q)G(\mathbb{Q})G(Q). Duke Math. J. 45 (1978), no. 4, 911-952.
[4] J. Arthur, A trace formula for reductive groups. II. Applications of a truncation operator. Compos. Math. 40 (1980), no. 1, 87-121.
[5] J. Arthur, A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983), no. 1-2, 1-89.
[6] J. Arthur, On elliptic tempered characters. Acta Math. 171 (1993), no. 1, 73-138.
[7] J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Amer. Math. Soc. Colloq. Publ. 61, American Mathematical Society, Providence, RI, 2013, xviii+590 pp.
[11] R. Beuzart-Plessis, On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad. Invent. Math. 214 (2018), no. 1, 437-521.
[13] R. Beuzart-Plessis, Comparison of local relative characters and the Ichino-Ikeda conjecture for unitary groups. J. Inst. Math. Jussieu 20 (2021), no. 6, 1803-1854.
[14] R. Beuzart-Plessis, A new proof of the Jacquet-Rallis fundamental lemma. Duke Math. J. 170 (2021), no. 12, 2805-2814.
[15] R. Beuzart-Plessis, Plancherel formula for GLn(F)∖GLn(E)GLn(F)∖GLn(E)GL_(n)(F)\\GL_(n)(E)\mathrm{GL}_{n}(F) \backslash \mathrm{GL}_{n}(E)GLn(F)∖GLn(E) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups. Invent. Math. 225 (2021), no. 1, 159-297.
[16] R. Beuzart-Plessis, P.-H. Chaudouard, and M. Zydor, The global Gan-GrossPrasad conjecture for unitary groups: the endoscopic case. Publ. Math. Inst. Hautes Études Sci. (2022).
[17] R. Beuzart-Plessis, Y. Liu, W. Zhang, and X. Zhu, Isolation of cuspidal spectrum, with application to the Gan-Gross-Prasad conjecture. Ann. of Math. (2) 194 (2021), no. 2, 519-584.
[18] R. Beuzart-Plessis and C. Wan, A local trace formula for the generalized Shalika model. Duke Math. J. 168 (2019), no. 7, 1303-1385.
[19] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G. Canad. J. Math. XLI (1989), no. 3, 385-438.
[20] P.-H. Chaudouard and M. Zydor, Le transfert singulier pour la formule des traces de Jacquet-Rallis. Compos. Math. 157 (2021), no. 2, 303-434.
[24] W. T. Gan and A. Ichino, The Gross-Prasad conjecture and local theta correspondence. Invent. Math. 206 (2016), no. 3, 705-799.
[25] D. Ginzburg, D. Jiang, and S. Rallis, On the nonvanishing of the central value of the Rankin-Selberg L-functions. J. Amer. Math. Soc. 17 (2004), no. 3, 679-722.
[26] B. H. Gross and D. Prasad, On irreducible representations of SO2n+1×SO2mSO2n+1×SO2mSO_(2n+1)xxSO_(2m)\mathrm{SO}_{2 n+1} \times \mathrm{SO}_{2 m}SO2n+1×SO2m. Canad. J. Math. 46 (1994), no. 5, 930-950.
[27] R. N. Harris, The refined Gross-Prasad conjecture for unitary groups. Int. Math. Res. Not. IMRN 2 (2014), 303-389.
[28] H. He, On the Gan-Gross-Prasad conjecture for U(p,q)U(p,q)U(p,q)U(p, q)U(p,q). Invent. Math. 209 (2017), no. 3, 837-884.
[29] K. Hiraga, A. Ichino, and T. Ikeda, Formal degrees and adjoint γγgamma\gammaγ-factors. J. Amer. Math. Soc. 21 (2008), no. 1, 283-304.
[30] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal. 19 (2010), no. 5, 1378-1425.
[31] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions. Amer. J. Math. 105 (1983), no. 2, 367-464.
[32] H. Jacquet and S. Rallis, On the Gross-Prasad conjecture for unitary groups. In On certain L-functions, pp. 205-264, Clay Math. Proc. 13, Amer. Math. Soc., Providence, RI, 2011.
[33] D. Jiang and L. Zhang, Arthur parameters and cuspidal automorphic modules of classical groups. Ann. of Math. (2) 191 (2020), no. 3, 739-827.
[34] T. Kaletha, A. Minguez, S. W. Shin, and P.-J. White, Endoscopic classification of representations: inner forms of unitary groups. 2014, arXiv:1409.3731.
[35] Y. Liu, Refined global Gan-Gross-Prasad conjecture for Bessel periods. J. Reine Angew. Math. 717 (2016), 133-194.
[36] Z. Luo, The local Gan-Gross-Prasad conjecture for special orthogonal groups over Archimedean local fields. 2021, arXiv:2102.11404.
[37] D. Mezer, Multiplicity one theorems over positive characteristic. 2020, arXiv:2010.16112.
[42] B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case. Ann. of Math. (2) 175 (2012), no. 1, 23-44.
[43] J. Tate, Number theoretic background. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 3-26, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, R.I., 1977.
[44] J. Tunnell, Local εεepsi\varepsilonε-factors and characters of GL(2). Amer. J. Math. 105 (1983), no. 6, 1277-1307.
[45] D. A. Vogan, The local Langlands conjecture. In Representation theory of groups and algebras, pp. 305-379, Contemp. Math. 145, Amer. Math. Soc., Providence, RI, 1993.
[50] C. Wan, A local relative trace formula for the Ginzburg-Rallis model: the geometric side. Mem. Amer. Math. Soc. 261 (2019), no. 1263.
[51] J. Xiao, Endoscopic transfer for unitary Lie algebras. 2018, arXiv:1802.07624
[52] H. Xue, The Gan-Gross-Prasad conjecture for U(n)×U(n)U(n)×U(n)U(n)xx U(n)U(n) \times U(n)U(n)×U(n). Adv. Math. 262 (2014), 1130-1191.
[53] H. Xue, On the global Gan-Gross-Prasad conjecture for unitary groups: approximating smooth transfer of Jacquet-Rallis. J. Reine Angew. Math. 756 (2019), 65−10065−10065-10065-10065−100.
[54] H. Xue, Bessel models for real unitary groups: the tempered case, 2020, https:// www.math.arizona.edu/ xuehang/lggp.pdf.
[55] Z. Yun, The fundamental lemma of Jacquet and Rallis. With an appendix by Julia Gordon. Duke Math. J. 156 (2011), no. 2, 167-227.
[56] W. Zhang, Automorphic period and the central value of Rankin-Selberg L-function. J. Amer. Math. Soc. 27 (2014), no. 2, 541-612.
[57] W. Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups. Ann. of Math. (2) 180 (2014), no. 3, 971-1049.
THE COHOMOLOGY OF SHIMURA VARIETIES WITH TORSION COEFFICIENTS
ANA CARAIANI
ABSTRACT
In this article, we survey recent work on some vanishing conjectures for the cohomology of Shimura varieties with torsion coefficients, under both local and global conditions. We discuss the ppppp-adic geometry of Shimura varieties and of the associated Hodge-Tate period morphism, and explain how this can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato-Tate conjecture for elliptic curves over CM fields.
Shimura varieties are algebraic varieties defined over number fields and equipped with many symmetries, which often provide a geometric realization of the Langlands correspondence. After base change to CCC\mathbb{C}C, they are closely related to certain locally symmetric spaces, but the beauty of Shimura varieties lies in their rich arithmetic.
To describe a Shimura variety, one needs to start with a Shimura datum (G,X)(G,X)(G,X)(G, X)(G,X). Here, GGGGG is a connected reductive group over QQQ\mathbb{Q}Q and XXXXX is a conjugacy class of homomorphisms h:ResC/RGm→GRh:ResC/Râ¡Gm→GRh:Res_(C//R)G_(m)rarrG_(R)h: \operatorname{Res}_{\mathbb{C} / \mathbb{R}} \mathbb{G}_{m} \rightarrow G_{\mathbb{R}}h:ResC/Râ¡Gm→GR of algebraic groups over RRR\mathbb{R}R. Both GGGGG and XXXXX are required to satisfy certain highly restrictive axioms, cf. [22, §2.1]. In particular, this allows one to give the conjugacy class XXXXX a more geometric flavor, as a variation of polarisable Hodge structures. One can show that such an XXXXX is a disjoint union of finitely many copies of Hermitian symmetric domains.
There is a complete classification of groups that admit a Shimura datum. For example, if G=GSp2nG=GSp2nG=GSp_(2n)G=\mathrm{GSp}_{2 n}G=GSp2n, one can take XXXXX to be the Siegel double space
(1.1){Z∈Mn(C)∣Z=Zt,Im(Z) positive or negative definite }(1.1)Z∈Mn(C)∣Z=Zt,Imâ¡(Z) positive or negative definite {:(1.1){Z inM_(n)(C)∣Z=Z^(t),Im(Z)" positive or negative definite "}:}\begin{equation*}
\left\{Z \in \mathrm{M}_{n}(\mathbb{C}) \mid Z=Z^{t}, \operatorname{Im}(Z) \text { positive or negative definite }\right\} \tag{1.1}
\end{equation*}(1.1){Z∈Mn(C)∣Z=Zt,Imâ¡(Z) positive or negative definite }
The associated Shimura varieties are called Siegel modular varieties and they are moduli spaces of principally polarized abelian varieties. Many other Shimura varieties - those of socalled "abelian type" - can be studied using moduli-theoretic techniques, by relating them to Siegel modular varieties. See [39] for an excellent introduction to the subject, which is focused on examples.
2. A VANISHING CONJECTURE FOR LOCALLY SYMMETRIC SPACES
Let G/QG/QG//QG / \mathbb{Q}G/Q be a connected reductive group. We consider the symmetric space associated with the Lie group G(R)G(R)G(R)G(\mathbb{R})G(R), which we define as X=G(R)/K∞∘A∞∘X=G(R)/K∞∘A∞∘X=G(R)//K_(oo)^(@)A_(oo)^(@)X=G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}X=G(R)/K∞∘A∞∘. Here, K∞∘K∞∘K_(oo)^(@)K_{\infty}^{\circ}K∞∘ is the connected component of the identity in a maximal compact subgroup K∞⊂G(R)K∞⊂G(R)K_(oo)sub G(R)K_{\infty} \subset G(\mathbb{R})K∞⊂G(R), and A∞∘A∞∘A_(oo)^(@)A_{\infty}^{\circ}A∞∘ is the connected component of the identity inside the real points of the maximal QQQ\mathbb{Q}Q-split torus in the center of GGGGG. Given a neat compact open subgroup K⊂G(Af)K⊂GAfK sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af), we can form the double quotient XK=G(Q)∖X×G(Af)/KXK=G(Q)∖X×GAf/KX_(K)=G(Q)\\X xx G(A_(f))//KX_{K}=G(\mathbb{Q}) \backslash X \times G\left(\mathbb{A}_{f}\right) / KXK=G(Q)∖X×G(Af)/K, which we call a locally symmetric space for GGGGG. This is a smooth Riemannian manifold, which does not have a complex structure, in general.
Example 2.1. If G=SL2/QG=SL2/QG=SL_(2)//QG=\mathrm{SL}_{2} / \mathbb{Q}G=SL2/Q, we can identify X=SL2(R)/SO2(R)X=SL2(R)/SO2(R)X=SL_(2)(R)//SO_(2)(R)X=\mathrm{SL}_{2}(\mathbb{R}) / \mathrm{SO}_{2}(\mathbb{R})X=SL2(R)/SO2(R) with the upper halfplane H2={z∈C∣Imz>0}H2={z∈C∣Imâ¡z>0}H^(2)={z inC∣Im z > 0}\mathbb{H}^{2}=\{z \in \mathbb{C} \mid \operatorname{Im} z>0\}H2={z∈C∣Imâ¡z>0} equipped with the hyperbolic metric, on which SL2(R)SL2(R)SL_(2)(R)\mathrm{SL}_{2}(\mathbb{R})SL2(R) acts transitively by the isometries
z↦az+bcz+d for z∈H2 and (abcd)∈SL2(R)z↦az+bcz+d for z∈H2 and abcd∈SL2(R)z|->(az+b)/(cz+d)" for "z inH^(2)quad" and "([a,b],[c,d])inSL_(2)(R)z \mapsto \frac{a z+b}{c z+d} \text { for } z \in \mathbb{H}^{2} \quad \text { and }\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \in \mathrm{SL}_{2}(\mathbb{R})z↦az+bcz+d for z∈H2 and (abcd)∈SL2(R)
In general, Shimura varieties are closely related to locally symmetric spaces, as in the first example, though the latter are much more general objects. For example, the locally symmetric spaces for G=ResF/QGLnG=ResF/Qâ¡GLnG=Res_(F//Q)GL_(n)G=\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}G=ResF/Qâ¡GLn do not arise from Shimura varieties if n≥3n≥3n >= 3n \geq 3n≥3, and, for n=2n=2n=2n=2n=2, they can only be related to Shimura varieties if FFFFF is a totally real field. In some instances, such as for ResF/QGL2ResF/Qâ¡GL2Res_(F//Q)GL_(2)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}ResF/Qâ¡GL2 with FFFFF a totally real field, one needs to replace G(R)/K∞∘A∞∘G(R)/K∞∘A∞∘G(R)//K_(oo)^(@)A_(oo)^(@)G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}G(R)/K∞∘A∞∘ by a slightly different quotient in order to obtain Shimura varieties. 11^(1){ }^{1}1 We now define the invariants
l0=rk(G(R))−rk(K∞)−rk(A∞) and q0=12(dimR(X)−l0)l0=rkâ¡(G(R))−rkâ¡K∞−rkâ¡A∞ and q0=12dimRâ¡(X)−l0l_(0)=rk(G(R))-rk(K_(oo))-rk(A_(oo))quad" and "quadq_(0)=(1)/(2)(dim_(R)(X)-l_(0))l_{0}=\operatorname{rk}(G(\mathbb{R}))-\operatorname{rk}\left(K_{\infty}\right)-\operatorname{rk}\left(A_{\infty}\right) \quad \text { and } \quad q_{0}=\frac{1}{2}\left(\operatorname{dim}_{\mathbb{R}}(X)-l_{0}\right)l0=rkâ¡(G(R))−rkâ¡(K∞)−rkâ¡(A∞) and q0=12(dimRâ¡(X)−l0)
These were first introduced by Borel-Wallach in [5]. There, they show up naturally in the computation of the ( g,K∞g,K∞g,K_(oo)\mathrm{g}, K_{\infty}g,K∞ )-cohomology of tempered representations of G(R)G(R)G(R)G(\mathbb{R})G(R). In the Shimura variety setting, we consider the variants l0=l0(Gad )l0=l0Gad l_(0)=l_(0)(G^("ad "))l_{0}=l_{0}\left(G^{\text {ad }}\right)l0=l0(Gad ) and q0=q0(Gad )q0=q0Gad q_(0)=q_(0)(G^("ad "))q_{0}=q_{0}\left(G^{\text {ad }}\right)q0=q0(Gad ) because of the different quotient used. In this case, l0(Gad )l0Gad l_(0)(G^("ad "))l_{0}\left(G^{\text {ad }}\right)l0(Gad ) can be shown to be equal to 0 by the second axiom in the definition of a Shimura datum.
As KKKKK varies, we have a tower of locally symmetric spaces (XK)KXKK(X_(K))_(K)\left(X_{K}\right)_{K}(XK)K, on which a spherical Hecke algebra TTT\mathbb{T}T for GGGGG acts by correspondences. The systems of Hecke eigenvalues occurring in the cohomology groups H(c)∗(XK,C)H(c)∗XK,CH_((c))^(**)(X_(K),C)H_{(c)}^{*}\left(X_{K}, \mathbb{C}\right)H(c)∗(XK,C) or, equivalently, the maximal ideals of TTT\mathbb{T}T in the support of these cohomology groups, can be related to automorphic representations of G(Af)GAfG(A_(f))G\left(\mathbb{A}_{f}\right)G(Af) by work of Franke and Matsushima [29]. The goal of this section is to state a conjecture on the cohomology of locally symmetric spaces with torsion coefficients FℓFâ„“F_(â„“)\mathbb{F}_{\ell}Fâ„“, where ℓâ„“â„“\ellâ„“ is a prime number. This conjecture is formulated in [25] (see the discussion around Conjecture 3.3) and in [12, CONJECTURE B]. Roughly, it says that the part of the cohomology outside the range of degrees [q0,q0+l0]q0,q0+l0[q_(0),q_(0)+l_(0)]\left[q_{0}, q_{0}+l_{0}\right][q0,q0+l0] is somehow degenerate. Note that this range of degrees is symmetric about the middle 12dimRX12dimRâ¡X(1)/(2)dim_(R)X\frac{1}{2} \operatorname{dim}_{\mathbb{R}} X12dimRâ¡X of the total range of cohomology and, in the Shimura variety case, it equals the middle degree of cohomology.
To formulate this more precisely, we use the notion of a non-Eisenstein maximal ideal in the Hecke algebra, for which we need to pass to the Galois side of the global Langlands correspondence. For simplicity, we will restrict our formulation to the case of G=G=G=G=G=ResF/QGLnResF/Qâ¡GLnRes_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Qâ¡GLn for some number field FFFFF, although the conjecture makes sense more generally. Let TTT\mathbb{T}T be the abstract spherical Hecke algebra away from a finite set SSSSS of primes of FFFFF and let m⊂Tm⊂TmsubT\mathfrak{m} \subset \mathbb{T}m⊂T be a maximal ideal in the support of H(c)∗(XK,Fℓ)H(c)∗XK,Fâ„“H_((c))^(**)(X_(K),F_(â„“))H_{(c)}^{*}\left(X_{K}, \mathbb{F}_{\ell}\right)H(c)∗(XK,Fâ„“). Assume that there exists a continuous, semisimple Galois representation ρ¯m:Gal(F¯/F)→GLn(F¯ℓ)ϯm:Galâ¡(F¯/F)→GLnF¯ℓbar(rho)_(m):Gal( bar(F)//F)rarrGL_(n)( bar(F)_(â„“))\bar{\rho}_{\mathfrak{m}}: \operatorname{Gal}(\bar{F} / F) \rightarrow \mathrm{GL}_{n}\left(\overline{\mathbb{F}}_{\ell}\right)ϯm:Galâ¡(F¯/F)→GLn(F¯ℓ) associated with mmm\mathfrak{m}m : by this, we mean that ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is unramified at all the primes of FFFFF away from the finite set SSSSS, and that, at any prime away from SSSSS, the Satake parameters of mmm\mathfrak{m}m match the Frobenius eigenvalues of ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm. (The precise condition is in terms of the characteristic polynomial of ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm applied to the Frobenius at such a prime and depends on various choices of normalizations. See, for example, [1, THEOREM 2.3.5] for a precise formulation.) Since the Galois representation is assumed to be semisimple and we are working with ResF/QGLnResF/Qâ¡GLnRes_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Qâ¡GLn, this property will characterize ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm by the Cebotarev density theorem and the Brauer-Nesbitt theorem. We say that mmm\mathfrak{m}m is non-Eisenstein if such a ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is absolutely irreducible.
The existence of ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm as above should be thought of as a modℓmodâ„“modâ„“\bmod \ellmodâ„“ version of the global Langlands correspondence, in the automorphic-to-Galois direction; in the case F=QF=QF=QF=\mathbb{Q}F=Q, this existence was conjectured by Ash [4]. The striking part of this conjecture is that it should apply to torsion classes in the cohomology of locally symmetric spaces, not just to those classes that lift to characteristic zero, and which can be related to automorphic representations of GGGGG. For general number fields, the existence of such Galois representations seems out of reach at the moment, even for classes in characteristic zero!
However, let FFFFF be a CM field: using nonstandard terminology, we mean that FFFFF is either a totally real field or a totally complex quadratic extension thereof. In this case, Scholze constructed such Galois representations in the breakthrough paper [53]. This strengthened previous work [33] that applied to cohomology with QℓQâ„“Q_(â„“)\mathbb{Q}_{\ell}Qâ„“-coefficients. Both these results relied, in turn, on the construction of Galois representations in the self-dual case, due to many people, including Clozel, Kottwitz, Harris-Taylor [34], Shin [61], and ChenevierHarris [21].
We can now state the promised vanishing conjecture for the cohomology of locally symmetric spaces with FℓFâ„“F_(â„“)\mathbb{F}_{\ell}Fâ„“-coefficients.
Conjecture 2.2. Assume that m⊂Tm⊂TmsubT\mathfrak{m} \subset \mathbb{T}m⊂T is a non-Eisenstein maximal ideal in the support of H(c)∗(XK,Fℓ)H(c)∗XK,Fâ„“H_((c))^(**)(X_(K),F_(â„“))H_{(c)}^{*}\left(X_{K}, \mathbb{F}_{\ell}\right)H(c)∗(XK,Fâ„“). Then H(c)i(XK,Fℓ)m≠0H(c)iXK,Fâ„“m≠0H_((c))^(i)(X_(K),F_(â„“))_(m)!=0H_{(c)}^{i}\left(X_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0H(c)i(XK,Fâ„“)m≠0 only if i∈[q0,q0+l0]i∈q0,q0+l0i in[q_(0),q_(0)+l_(0)]i \in\left[q_{0}, q_{0}+l_{0}\right]i∈[q0,q0+l0].
where χχchi\chiχ is a suitable modℓmodâ„“modâ„“\bmod \ellmodâ„“ character of Gal(Q¯/Q)Galâ¡(Q¯/Q)Gal( bar(Q)//Q)\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})Galâ¡(Q¯/Q) and χcyclo :Gal(Q¯/Q)→Fℓ×χcyclo :Galâ¡(Q¯/Q)→Fℓ×chi_("cyclo "):Gal( bar(Q)//Q)rarrF_(â„“)^(xx)\chi_{\text {cyclo }}: \operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \rightarrow \mathbb{F}_{\ell}^{\times}χcyclo :Galâ¡(Q¯/Q)→Fℓ×is the modℓmodâ„“modâ„“\bmod \ellmodâ„“ cyclotomic character. Later, we will introduce a local genericity condition at an auxiliary prime p≠ℓp≠ℓp!=â„“p \neq \ellp≠ℓ and we will see that the ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm in (2.1) also fail to satisfy genericity everywhere. In addition to these and a few more low-dimensional examples, one can also consider the analogue of Conjecture 2.2 for H(c)∗(XK,Qℓ)H(c)∗XK,Qâ„“H_((c))^(**)(X_(K),Q_(â„“))H_{(c)}^{*}\left(X_{K}, \mathbb{Q}_{\ell}\right)H(c)∗(XK,Qâ„“). This analogue is related to Arthur's conjectures on the cohomology of locally symmetric spaces [3] and can be verified for GLnGLnGL_(n)\mathrm{GL}_{n}GLn over CMCMCM\mathrm{CM}CM fields using work of Franke and Borel-Wallach (see [1, theOREM 2.4.9]).
Conjecture 2.2 is motivated by the Calegari-Geraghty enhancement [12] of the classical Taylor-Wiles method for proving automorphy lifting theorems. The classical method works well in settings where the (co)homology of locally symmetric spaces is concentrated in one degree, for example, for GL2/QGL2/QGL_(2)//Q\mathrm{GL}_{2} / \mathbb{Q}GL2/Q after localizing at a non-Eisenstein maximal ideal, or for definite unitary groups over totally real fields. In general, however, a certain numerical coincidence that is used to compare the Galois and automorphic sides breaks down. Calegari and Geraghty had a significant insight: they reinterpret the failure of the numerical coincidence in terms of the invariant l0l0l_(0)l_{0}l0. More precisely, l0l0l_(0)l_{0}l0 arises naturally from a computation on the Galois side, and the commutative algebra underlying the method can be adjusted if one knows that the cohomology on the automorphic side, after localizing at a non-Eisenstein maximal ideal, is concentrated in a range of degrees of length at most l0l0l_(0)l_{0}l0. For an overview of the key ideas involved in the Calegari-Geraghty method, see [10, §10].
In the case of Shimura varieties, Conjecture 2.2 predicts that the non-Eisenstein part of the cohomology with FℓFâ„“F_(â„“)\mathbb{F}_{\ell}Fâ„“-coefficients is concentrated in the middle degree. The initial progress on this conjecture in the Shimura variety setting had rather strong additional assumptions: for example, one needed ℓâ„“â„“\ellâ„“ to be an unramified prime for the Shimura datum and KℓKâ„“K_(â„“)K_{\ell}Kâ„“ to be hyperspecial, as in the work of Dimitrov [23] and Lan-Suh [40,41]. The theory of perfectoid Shimura varieties and their associated Hodge-Tate period morphism has been a game-changer in this area. For the rest of this article, we will discuss more recent progress on Conjecture 2.2 and related questions in the special case of Shimura varieties, as well as applications that go beyond the setting of Shimura varieties.
3. THE HODGE-TATE PERIOD MORPHISM
The Hodge-Tate period morphism was introduced by Scholze in his breakthrough paper [53] and it was subsequently refined in [17]. It gives an entirely new way to think about the geometry and cohomology of Shimura varieties. In the past decade, it had numerous striking applications to the Langlands programme: to Scholze's construction of Galois representations for torsion classes, to the vanishing theorems discussed in Sections 4 and 5, to the construction of higher Coleman theory by Boxer and Pilloni [8], and to a radically new approach to the Fontaine-Mazur conjecture due to Pan [48].
For simplicity, let us consider a Shimura datum (G,X)(G,X)(G,X)(G, X)(G,X) of of Hodge type. By this, we mean that (G,X)(G,X)(G,X)(G, X)(G,X) admits a closed embedding into a Siegel datum (G~,X~)(G~,X~)( tilde(G), tilde(X))(\tilde{G}, \tilde{X})(G~,X~), where G~=GSp2nG~=GSp2ntilde(G)=GSp_(2n)\tilde{G}=\mathrm{GSp}_{2 n}G~=GSp2n, for some n∈Z≥1n∈Z≥1n inZ_( >= 1)n \in \mathbb{Z}_{\geq 1}n∈Z≥1, and X~X~tilde(X)\tilde{X}X~ is as in (1.1). For example, (G,X)(G,X)(G,X)(G, X)(G,X) could be a Shimura datum of PELPELPELP E LPEL type arising from a unitary similitude group: the corresponding Shimura varieties will represent a moduli problem of abelian varieties equipped with extra structures (polarizations, endomorphisms, and level structures). This unitary case will be the main example to keep in mind, as this will also play a central role in Section 4.
For some representative h∈Xh∈Xh in Xh \in Xh∈X, we consider the Hodge cocharacter
The axioms in the definition of the Shimura datum imply that μμmu\muμ is minuscule. The reflex field EEEEE is the field of definition of the conjugacy class {μ}{μ}{mu}\{\mu\}{μ}; it is a finite extension of QQQ\mathbb{Q}Q and the corresponding Shimura varieties admit canonical models over EEEEE. The cocharacter μμmu\muμ also determines two opposite parabolic subgroups Pμstd Pμstd P_(mu)^("std ")P_{\mu}^{\text {std }}Pμstd and PμPμP_(mu)P_{\mu}Pμ, whose conjugacy classes are defined over EEEEE. These are given by
Pμstd ={g∈G∣limt→∞ad(μ(t))g exists },Pμ={g∈G∣limt→0ad(μ(t))g exists }Pμstd =g∈G∣limt→∞ adâ¡(μ(t))g exists ,Pμ=g∈G∣limt→0 adâ¡(μ(t))g exists P_(mu)^("std ")={g in G∣lim_(t rarr oo)ad(mu(t))g" exists "},quadP^(mu)={g in G∣lim_(t rarr0)ad(mu(t))g" exists "}P_{\mu}^{\text {std }}=\left\{g \in G \mid \lim _{t \rightarrow \infty} \operatorname{ad}(\mu(t)) g \text { exists }\right\}, \quad P^{\mu}=\left\{g \in G \mid \lim _{t \rightarrow 0} \operatorname{ad}(\mu(t)) g \text { exists }\right\}Pμstd ={g∈G∣limt→∞adâ¡(μ(t))g exists },Pμ={g∈G∣limt→0adâ¡(μ(t))g exists }
We let Flstd Flstd Fl^("std ")\mathrm{Fl}^{\text {std }}Flstd and FlFlFl\mathrm{Fl}Fl denote the associated flag varieties, which are also defined over EEEEE.
(3.1)πdR:X↪Flstd(C)=G(C)/Pμstd(3.1)Ï€dR:X↪Flstd(C)=G(C)/Pμstd{:(3.1)pi_(dR):X↪Fl^(std)(C)=G(C)//P_(mu)^(std):}\begin{equation*}
\pi_{\mathrm{dR}}: X \hookrightarrow \mathrm{Fl}^{\mathrm{std}}(\mathbb{C})=G(\mathbb{C}) / P_{\mu}^{\mathrm{std}} \tag{3.1}
\end{equation*}(3.1)πdR:X↪Flstd(C)=G(C)/Pμstd
called the Borel embedding, defined by h↦Fil∙(μh)h↦Fil∙μhh|->Fil^(∙)(mu_(h))h \mapsto \mathrm{Fil}^{\bullet}\left(\mu_{h}\right)h↦Fil∙(μh). The axioms of a Shimura datum imply that XXXXX is a variation of polarisable Hodge structures of abelian varieties. Modulitheoretically, πdRÏ€dRpi_(dR)\pi_{\mathrm{dR}}Ï€dR sends a Hodge structure, such as
On the other hand, the parabolic subgroup PμPμP_(mu)P_{\mu}Pμ is the stabilizer of the ascending filtration Fil. This gives rise to an antiholomorphic embedding
Let ppppp be a rational prime, p∣pp∣pp∣p\mathfrak{p} \mid pp∣p a prime of EEEEE, and let CCCCC be the completion of an algebraic closure of EpEpE_(p)E_{\mathfrak{p}}Ep. We consider the adic spaces ∮K∮Koint_(K)\oint_{K}∮K and FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“ over Spa(C,OC)Spaâ¡C,OCSpa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spaâ¡(C,OC) corresponding to the algebraic varieties SKSKS_(K)S_{K}SK and FlFlFl\mathrm{Fl}Fl over EEEEE. A striking result of Scholze shows that the tower of Shimura varieties (SKpKp)KpSKpKpKp(S_(K^(p)K_(p)))_(K_(p))\left(S_{K^{p} K_{p}}\right)_{K_{p}}(SKpKp)Kp acquires the structure of a perfectoid space (in the sense of [51]) as KpKpK_(p)K_{p}Kp varies over compact open subgroups of G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). More precisely, the following result was established in [53,$3,4][53,$3,4][53,$3,4][53, \$ 3,4][53,$3,4] and later refined in [17,$2][17,$2][17,$2][17, \$ 2][17,$2], by correctly identifying the target of the Hodge-Tate period morphism.
Theorem 3.1. There exists a unique perfectoid space SKpSKpS_(K^(p))S_{K^{p}}SKp satisfying SKp∼limKpSKpKp,2SKp∼limKp SKpKp,2S_(K^(p))∼_(lim_(K_(p)))S_(K^(p)K_(p)),^(2)S_{K^{p}} \sim \underset{\lim _{K_{p}}}{ } \mathcal{S}_{K^{p} K_{p}},{ }^{2}SKp∼limKpSKpKp,2 in the sense of [55, DEFINITION 2.4.1], and a G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp)-equivariant morphism of adic spaces
Moreover, πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT is equivariant for the usual action of Hecke operators away from ppppp on SKpSKpS_(K^(p))S_{K^{p}}SKp and their trivial action on FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“.
In the Siegel case G=GSp2n/QG=GSp2n/QG=GSp_(2n)//QG=\mathrm{GSp}_{2 n} / \mathbb{Q}G=GSp2n/Q, one can describe the Hodge-Tate period morphism πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT from a moduli-theoretic perspective as follows. An abelian variety A/CA/CA//CA / CA/C, equipped with a trivialization TpA≃Zp2nTpA≃Zp2nT_(p)A≃Z_(p)^(2n)T_{p} A \simeq \mathbb{Z}_{p}^{2 n}TpA≃Zp2n will be sent to the first piece of the Hodge-Tate filtration
Lie A⊂TpA⊗ZpC≃C2nA⊂TpA⊗ZpC≃C2nA subT_(p)Aox_(Z_(p))C≃C^(2n)A \subset T_{p} A \otimes_{\mathbb{Z}_{p}} C \simeq C^{2 n}A⊂TpA⊗ZpC≃C2n.
where (−1)(−1)(-1)(-1)(−1) denotes a Tate twist (which is important for keeping track of the Galois action). To show that the morphism defined this way on Spa(C,C+)Spaâ¡C,C+Spa(C,C^(+))\operatorname{Spa}\left(C, C^{+}\right)Spaâ¡(C,C+)-points comes from a morphism of adic spaces, it is important to know that the filtration (3.3) varies continuously. At the same time, to extend the result to Shimura varieties of Hodge type and to cut down the image to FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“, one needs to keep track of Hodge tensors carefully. Both problems are solved via relative ppppp-adic Hodge theory for the morphism AK→SKAK→SKA_(K)rarrS_(K)\mathscr{A}_{K} \rightarrow S_{K}AK→SK, where AKAKA_(K)\mathcal{A}_{K}AK is the restriction to ςKÏ‚KÏ‚_(K)\varsigma_{K}Ï‚K of a universal abelian scheme over an ambient Siegel modular variety. See [13, §3] for an overview.
Theorem 3.1 can be extended to minimal and toroidal compactifications of Siegel modular varieties, cf. [53] and [49]. Moreover, there is a natural affinoid cover of FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“ such that the preimage under πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT of each affinoid in the cover is an affinoid perfectoid subspace of SKp∗SKp∗S_(K^(p))^(**)S_{K^{p}}^{*}SKp∗. The underlying reason for this is the fact that the partial minimal compactification of the ordinary locus is affine. The perfectoid structure on SKp∗SKp∗S_(K^(p))^(**)S_{K^{p}}^{*}SKp∗ and the affinoid nature of the Hodge-Tate period morphism play an important role in Scholze's ppppp-adic interpolation argument, that is key for the construction of Galois representations associated with torsion classes. See also [44] for an exposition of the main ideas.
Theorem 3.1 can also be extended to minimal and toroidal compactifications of Shimura varieties of Hodge type and even abelian type, cf. [32, 58] and [8], although there are some technical issues at the boundary. For example, the cleanest formulation currently available in full generality is that the relationship SKp∗=lim⟵KpSKpKp∗SKp∗=lim⟵Kp SKpKp∗S_(K^(p))^(**)=lim_(longleftarrowK_(p))S_(K^(p)K_(p))^(**)S_{K^{p}}^{*}=\lim _{\longleftarrow K_{p}} S_{K^{p} K_{p}}^{*}SKp∗=lim⟵KpSKpKp∗, for a perfectoid space SKp∗SKp∗S_(K^(p))^(**)\mathcal{S}_{K^{p}}^{*}SKp∗, holds in Scholze's category of diamonds [54].
Example 3.2. To see where the perfectoid structure on SKpSKpS_(K^(p))S_{K^{p}}SKp comes from, it is instructive to consider the case of modular curves and study the geometry of their special fibers: we are particularly interested in the geometry of the so-called Deligne-Rapoport model. Set G=G=G=G=G=GL2/QGL2/QGL_(2)//Q\mathrm{GL}_{2} / \mathbb{Q}GL2/Q. Let Kp0=GL2(Zp)Kp0=GL2ZpK_(p)^(0)=GL_(2)(Z_(p))K_{p}^{0}=\mathrm{GL}_{2}\left(\mathbb{Z}_{p}\right)Kp0=GL2(Zp), the hyperspecial compact open subgroup and let S¯KpKp0/FpS¯KpKp0/Fpbar(S)_(K^(p)K_(p)^(0))//F_(p)\bar{S}_{K^{p} K_{p}^{0}} / \mathbb{F}_{p}S¯KpKp0/Fp be the special fiber of the integral model over Z(p)Z(p)Z_((p))\mathbb{Z}_{(p)}Z(p) of the modular curve at this level. This is a smooth curve over FpFpF_(p)\mathbb{F}_{p}Fp that represents a moduli problem (E,α)(E,α)(E,alpha)(E, \alpha)(E,α) of elliptic curves equipped with prime-to- ppppp level structures (determined by the prime-to- ppppp level KpKpK^(p)K^{p}Kp ). The isogeny class of the ppppp-divisible group E[p∞]Ep∞E[p^(oo)]E\left[p^{\infty}\right]E[p∞] induces the Newton stratification
into an open dense ordinary stratum S¯KpKp0ord S¯KpKp0ord bar(S)_(K^(p)K_(p)^(0))^("ord ")\bar{S}_{K^{p} K_{p}^{0}}^{\text {ord }}S¯KpKp0ord (where E[p∞]Ep∞E[p^(oo)]E\left[p^{\infty}\right]E[p∞] is isogenous to μp∞×Qp/Zpμp∞×Qp/Zpmu_(p^(oo))xxQ_(p)//Z_(p)\mu_{p^{\infty}} \times \mathbb{Q}_{p} / \mathbb{Z}_{p}μp∞×Qp/Zp ) and a closed supersingular stratum S¯KpssKp0S¯KpssKp0bar(S)_(K^(p))^(ss)K_(p)^(0)\bar{S}_{K^{p}}^{\mathrm{ss}} K_{p}^{0}S¯KpssKp0 consisting of finitely many points (where E[p∞]Ep∞E[p^(oo)]E\left[p^{\infty}\right]E[p∞] is connected).
Now let Kp1⊂GL2(Qp)Kp1⊂GL2QpK_(p)^(1)subGL_(2)(Q_(p))K_{p}^{1} \subset \mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)Kp1⊂GL2(Qp) be the Iwahori subgroup and S¯KpKp1/FpS¯KpKp1/Fpbar(S)_(K^(p)K_(p)^(1))//F_(p)\bar{S}_{K^{p} K_{p}^{1}} / \mathbb{F}_{p}S¯KpKp1/Fp be the special fiber of the integral model of the modular curve at this level. This represents a moduli problem (E,α,D)(E,α,D)(E,alpha,D)(E, \alpha, D)(E,α,D) of elliptic curves equipped with prime-to- ppppp level structures and also with a level structure at ppppp given by a finite flat subgroup scheme D⊂E[p]D⊂E[p]D sub E[p]D \subset E[p]D⊂E[p] of order ppppp. Again, we have the
preimage of the Newton stratification S¯KpKp1=S¯KpKp1ord ⊔S¯KpKp1ssS¯KpKp1=S¯KpKp1ord ⊔S¯KpKp1ssbar(S)_(K^(p)K_(p)^(1))= bar(S)_(K^(p)K_(p)^(1))^("ord ")⊔ bar(S)_(K^(p)K_(p)^(1))^(ss)\bar{S}_{K^{p} K_{p}^{1}}=\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }} \sqcup \bar{S}_{K^{p} K_{p}^{1}}^{\mathrm{ss}}S¯KpKp1=S¯KpKp1ord ⊔S¯KpKp1ss. The modular curve at this level is not smooth, but rather a union of irreducible components that intersect transversely at the finitely many supersingular points.
The open and dense ordinary locus S¯KpKp1ord S¯KpKp1ord bar(S)_(K^(p)K_(p)^(1))^("ord ")\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }}S¯KpKp1ord is a disjoint union of two KottwitzRapoport strata: the one where D≃μpD≃μpD≃mu_(p)D \simeq \mu_{p}D≃μp and the one where D≃Z/pZD≃Z/pZD≃Z//pZD \simeq \mathbb{Z} / p \mathbb{Z}D≃Z/pZ. Both of these Kottwitz-Rapoport strata can be shown to be abstractly isomorphic to the ordinary stratum at hyperspecial level. If we restrict the natural forgetful map S¯KpKp1ord →S¯KpKp0ord S¯KpKp1ord →S¯KpKp0ord bar(S)_(K^(p)K_(p)^(1))^("ord ")rarr bar(S)_(K^(p)K_(p)^(0))^("ord ")\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }} \rightarrow \bar{S}_{K^{p} K_{p}^{0}}^{\text {ord }}S¯KpKp1ord →S¯KpKp0ord to the Kottwitz-Rapoport stratum where D≃Z/pZD≃Z/pZD≃Z//pZD \simeq \mathbb{Z} / p \mathbb{Z}D≃Z/pZ, the map can be identified (up to an isomorphism) with the geometric Frobenius. (The restriction of the map to the Kottwitz-Rapoport stratum where D≃μpD≃μpD≃mu_(p)D \simeq \mu_{p}D≃μp is an isomorphism.)
On the adic generic fiber, one can extend this picture to an anticanonical ordinary tower, where the transition morphisms reduce modulo ppppp to (powers of) the geometric Frobenius, giving a perfectoid space in the limit. To extend beyond the ordinary locus, Scholze uses the theory of the canonical subgroup, the action of GL2(Qp)GL2QpGL_(2)(Q_(p))\mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)GL2(Qp) at infinite level, and a rudimentary form of the Hodge-Tate period morphism that is just defined on the underlying topological spaces.
The above strategy generalizes relatively cleanly to higher-dimensional Siegel modular varieties, modulo subtleties at the boundary. To extend Theorem 3.1 to general Shimura varieties of Hodge type, Scholze considers an embedding at infinite level into a Siegel modular variety. It is surprisingly subtle to understand directly the perfectoid structure on a general Shimura variety of Hodge type (especially in the case when GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is nonsplit) and this is related to the discussion in Section 5. This is also related to the fact that the geometry of the EKOR stratification is more intricate when GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is nonsplit.
For simplicity, let us now assume that (G,X)(G,X)(G,X)(G, X)(G,X) is a Shimura datum of PEL type and that ppppp is an unramified prime for this Shimura datum. Recall the Kottwitz set B(G)B(G)B(G)B(G)B(G) classifying isocrystals with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure. The Hodge cocharacter μμmu\muμ defines a subset B(G,μ−1)⊂BG,μ−1⊂B(G,mu^(-1))subB\left(G, \mu^{-1}\right) \subsetB(G,μ−1)⊂B(G)B(G)B(G)B(G)B(G) of μ−1μ−1mu^(-1)\mu^{-1}μ−1-admissible elements. The special fiber of the Shimura variety with hyperspecial level at ppppp admits a Newton stratification
S¯KpKp0=⨆b∈B(G,μ−1)S¯KpKp0bS¯KpKp0=⨆b∈BG,μ−1 S¯KpKp0bbar(S)_(K^(p)K_(p)^(0))=⨆_(b in B(G,mu^(-1))) bar(S)_(K^(p)K_(p)^(0))^(b)\bar{S}_{K^{p} K_{p}^{0}}=\bigsqcup_{b \in B\left(G, \mu^{-1}\right)} \bar{S}_{K^{p} K_{p}^{0}}^{b}S¯KpKp0=⨆b∈B(G,μ−1)S¯KpKp0b
into locally closed strata indexed by this subset. This stratification is in terms of isogeny classes of ppppp-divisible groups with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure and generalizes the stratification (3.4) from the modular curve case.
For each b∈B(G,μ−1)b∈BG,μ−1b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can choose a (completely slope divisible) ppppp-divisible group with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure Xb/F¯pXb/F¯pX_(b)// bar(F)_(p)\mathbb{X}_{b} / \overline{\mathbb{F}}_{p}Xb/F¯p and define the corresponding Oort central leaf. This is a
smooth closed subscheme CXbCXbC^(Xb)\mathscr{C}^{\mathbb{X} b}CXb of the Newton stratum S¯KpKp0bS¯KpKp0bbar(S)_(K^(p)K_(p)^(0))^(b)\bar{S}_{K^{p} K_{p}^{0}}^{b}S¯KpKp0b, such that the isomorphism class of the ppppp-divisible group with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure over each geometric point of the leaf is constant and equal to that of XbXbX_(b)\mathbb{X}_{b}Xb :
In general, there can be infinitely many leaves inside a given Newton stratum. Over each central leaf, one has the perfect Igusa variety Igb/F¯pIgbâ¡/F¯pIg^(b)// bar(F)_(p)\operatorname{Ig}^{b} / \bar{F}_{p}Igbâ¡/F¯p, a profinite cover of CXbCXbC^(X^(b))\mathscr{C}^{\mathbb{X}^{b}}CXb which parametrizes trivializations of the universal ppppp-divisible group with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure.
While the central leaf CXbCXbC^(X_(b))\mathscr{C}^{\mathbb{X}_{b}}CXb depends on the choice of XbXbX_(b)\mathbb{X}_{b}Xb in its isogeny class, one can show that the perfect Igusa variety IgbIgbIg^(b)\mathrm{Ig}^{b}Igb only depends on the isogeny class: this follows from the equivalent moduli-theoretic description in [17, LEMMA 4.3.4] (see also [19, LEMMA 4.2.2], which keeps track of the extra structures more carefully). In particular, the pair (G,μ)(G,μ)(G,mu)(G, \mu)(G,μ) is not determined by the Igusa variety IgbIgbIg^(b)\mathrm{Ig}^{b}Igb - it can happen that Igusa varieties that are a priori obtained from different Shimura varieties are isomorphic. See [19, THEOREM 4.2.4] for an example and [57] for a systematic analysis of this phenomenon in the function field setting.
Because Igb/F¯pIgb/F¯pIg^(b)// bar(F)_(p)\mathrm{Ig}^{b} / \overline{\mathbb{F}}_{p}Igb/F¯p is perfect, the base change Igb×F¯pOC/pIgb×F¯pOC/pIg^(b)xx bar(F)_(p)O_(C)//p\mathrm{Ig}^{b} \times \overline{\mathbb{F}}_{p} \mathcal{O}_{C} / pIgb×F¯pOC/p admits a canonical lift to a flat formal scheme over SpfOCSpfâ¡OCSpf O_(C)\operatorname{Spf} \mathcal{O}_{C}Spfâ¡OC. We let ℑgbâ„‘gbâ„‘g^(b)\mathfrak{\Im} \mathfrak{g}^{b}â„‘gb denote the adic generic fiber of this lift, which is a perfectoid space over Spa(C,OC)Spaâ¡C,OCSpa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spaâ¡(C,OC). The spaces IgbIgbIg^(b)\mathrm{Ig}^{b}Igb and ℑgbâ„‘gbâ„‘g^(b)\mathfrak{\Im} \mathfrak{g}^{b}â„‘gb have naturally isomorphic ℓâ„“â„“\ellâ„“-adic cohomology groups and they both have an action of a locally profinite group Gb(Qp)GbQpG_(b)(Q_(p))G_{b}\left(\mathbb{Q}_{p}\right)Gb(Qp), where GbGbG_(b)G_{b}Gb is an inner form of a Levi subgroup of GGGGG.
For each b∈B(G,μ−1)b∈BG,μ−1b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can also consider the associated Rapoport-Zink space, a moduli space of ppppp-divisible groups with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure that is a local analogue of a Shimura variety. Concretely in the PEL case, one considers a moduli problem of ppppp-divisible groups equipped with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure, satisfying the Kottwitz determinant condition with respect to μμmu\muμ, and with a modulo ppppp quasiisogeny to the fixed ppppp-divisible group XbXbX_(b)\mathbb{X}_{b}Xb. This moduli problem was shown by Rapoport-Zink [50] to be representable by a formal scheme over SpfOE˘pSpfâ¡OE˘pSpf O_(E^(˘)_(p))\operatorname{Spf} \mathcal{O}_{\breve{E}_{\mathfrak{p}}}Spfâ¡OE˘p, where E˘pE˘pE^(˘)_(p)\breve{E}_{\mathfrak{p}}E˘p is the completion of the maximal unramified extension of EpEpE_(p)E_{\mathfrak{p}}Ep. We let MbMbM^(b)\mathcal{M}^{b}Mb denote the adic generic fiber of this formal scheme, 33^(3){ }^{3}3 base changed to Spa(C,OC)Spaâ¡C,OCSpa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spaâ¡(C,OC), and let M∞bM∞bM_(oo)^(b)\mathcal{M}_{\infty}^{b}M∞b denote the corresponding infinite-level Rapoport-Zink space. The latter object can be shown to be a perfectoid space using the techniques of [55], by which the infinite-level Rapoport-Zink space admits a local analogue of the Hodge-Tate period morphism
It turns out that the geometry of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT is intricately tied up with the geometry of its local analogues πHTbÏ€HTbpi_(HT)^(b)\pi_{\mathrm{HT}}^{b}Ï€HTb. The following result is a conceptually cleaner, infinite-level version of the Mantovan product formula established in [43], which describes Newton strata inside Shimura varieties in terms of a product of Igusa varieties and Rapoport-Zink spaces.
Theorem 3.3. There exists a Newton stratification
Fℓ=⨆b∈B(G,μ−1)FℓbFâ„“=⨆b∈BG,μ−1 Fâ„“bFâ„“=⨆_(b in B(G,mu^(-1)))Fâ„“^(b)\mathscr{F} \ell=\bigsqcup_{b \in B\left(G, \mu^{-1}\right)} \mathscr{F} \ell^{b}Fâ„“=⨆b∈B(G,μ−1)Fâ„“b
into locally closed strata.
For each b∈B(G,μ−1)b∈BG,μ−1b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can consider the Newton stratum ∮Kp∘b∮Kp∘boint_(K^(p))^(@b)\oint_{K^{p}}^{\circ b}∮Kp∘b as a locally closed subspace of the good reduction locus SKp∘SKp∘S_(K^(p))^(@)S_{K^{p}}^{\circ}SKp∘. There exists a Cartesian diagram of diamonds over Spd(C,OC)Spdâ¡C,OCSpd(C,O_(C))\operatorname{Spd}\left(C, \mathcal{O}_{C}\right)Spdâ¡(C,OC)
The decomposition into Newton strata is defined in [17, §3]. Morally, one first constructs a map of v-stacks Fℓ→Fℓ→Fâ„“rarr\mathscr{F} \ell \rightarrowFℓ→ Bun GG_(G)_{G}G, where the latter is the vvvvv-stack of GGGGG-bundles on the Fargues-Fontaine curve. To construct this map of v-stacks, it is convenient to notice that one can identify the diamond associated to FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“ with the minuscule Schubert cell defined by μμmu\muμ inside the BdR+BdR+B_(dR)^(+)B_{\mathrm{dR}}^{+}BdR+-Grassmannian for GGGGG. Once the map to BunGBunGBun_(G)\operatorname{Bun}_{G}BunG is in the picture, one uses Fargues's result that the points of Bun GG_(G){ }_{G}G are in bijection with the Kottwitz set B(G)B(G)B(G)B(G)B(G), cf. [27] (see also [2] for an alternative proof that also works in equal characteristic). Moreover, the Newton decomposition is a stratification, in the sense that, for b∈B(G,μ)b∈B(G,μ)b in B(G,mu)b \in B(G, \mu)b∈B(G,μ), we have
where ≥≥>=\geq≥ denotes the Bruhat order. The latter fact follows from a recent result of Viehmann, see [63, THEOREM 1.1].
On rank one points, πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT is compatible with the two different ways of defining the Newton stratification: via pullback from S¯KpKp0S¯KpKp0bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0 on §Kp§Kp§_(K^(p))\S_{K^{p}}§§Kp and via pullback from Bun GG_(G){ }_{G}G on FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“. The behavior is more subtle on higher rank points. This is related to the fact that the closure relations are reversed in the two settings: the basic locus inside S¯KpKp0S¯KpKp0bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0 is the unique closed stratum, whereas each basic stratum inside BunGBunGBun_(G)\mathrm{Bun}_{G}BunG is open. On the other hand, the (μ)(μ)(mu)(\mu)(μ)-ordinary locus is open and dense inside S¯KpKp0S¯KpKp0bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0, whereas it is a zero-dimensional closed stratum inside FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“. The infinite-level product formula is established in [17, $4], although it is formulated in terms of functors on PerfE˘p⋅4PerfE˘pâ‹…4Perf_(E^(˘)_(p))*^(4)\operatorname{Perf}_{\breve{E}_{\mathfrak{p}}} \cdot{ }^{4}PerfE˘pâ‹…4 This was extended to Shimura varieties of Hodge type by Hamacher [31].
Assume that the Shimura varieties SKSKS_(K)S_{K}SK are compact. We have the following consequence for the fibers of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT : let x¯:Spa(C,C+)→Fℓbx¯:SpaC,C+→Fâ„“bbar(x):Spa(C,C^(+))rarrFâ„“^(b)\bar{x}: \mathrm{Spa}\left(C, C^{+}\right) \rightarrow \mathscr{F} \ell^{b}x¯:Spa(C,C+)→Fâ„“b be a geometric point. Then there is an inclusion of ℑgbâ„‘gbâ„‘g^(b)\mathfrak{\Im g}^{b}â„‘gb into πHT−1(x¯)Ï€HT−1(x¯)pi_(HT)^(-1)( bar(x))\pi_{\mathrm{HT}}^{-1}(\bar{x})Ï€HT−1(x¯), which identifies the target with the canonical compactification of the source, in the sense of [54, PROPOSITION 18.6]. In [18, THEOREM 1.10], we extend the computation of the fibers to minimal and toroidal compactifications of (noncompact) Shimura varieties attached to quasisplit unitary groups. In this case, the fibers can be obtained from partial minimal and toroidal compactifications of Igusa varieties. It would be interesting to extend the whole infinite-level product formula to compactifications.
Example 3.4. We make the geometry of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT explicit in the case of the modular curve, i.e., for G=GL2/QG=GL2/QG=GL_(2)//QG=\mathrm{GL}_{2} / \mathbb{Q}G=GL2/Q. In this case, we identify Fℓ=P1, ad Fâ„“=P1, ad Fâ„“=P^(1," ad ")\mathscr{F} \ell=\mathbb{P}^{1, \text { ad }}Fâ„“=P1, ad and we have the decomposition into Newton strata
The fibers of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT over the ordinary locus are "perfectoid versions" of Igusa curves. The infinite-level version of the product formula reduces, in this case, to the statement that the ordinary locus is parabolically induced from ℑgord â„‘gord ℑg^("ord ")\mathfrak{\Im} \mathrm{g}^{\text {ord }}â„‘gord , as in [19,$6][19,$6][19,$6][19, \$ 6][19,$6]. The fibers of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT over the supersingular locus are profinite sets: the corresponding Igusa varieties can be identified with double cosets D×∖D×(Afp)/KpD×∖D×Afp/KpD^(xx)\\D^(xx)(A_(f)^(p))//K^(p)D^{\times} \backslash D^{\times}\left(\mathbb{A}_{f}^{p}\right) / K^{p}D×∖D×(Afp)/Kp, where D/QD/QD//QD / \mathbb{Q}D/Q is the quaternion algebra ramified precisely at ∞∞oo\infty∞ and ppppp. This precise result is established in [35], although the idea goes back to DeuringSerre. One should be able to give an analogous description for basic Igusa varieties in much greater generality - this is closely related to Rapoport-Zink uniformization.
4. COHOMOLOGY WITH MOD ℓâ„“â„“\ellâ„“ COEFFICIENTS
In this section, we outline some recent strategies for computing the cohomology of Shimura varieties with modulo ℓâ„“â„“\ellâ„“ coefficients using the ppppp-adic Hodge-Tate period morphism, where ℓâ„“â„“\ellâ„“ and ppppp are two distinct primes. We emphasize the strategies developed in [17-19,38].
We will assume throughout that (G,X)(G,X)(G,X)(G, X)(G,X) is a Shimura datum of abelian type and, in practice, we will focus on two examples: the case of Shimura varieties associated with unitary similitude groups and the case of Hilbert modular varieties. Let m⊂Tm⊂TmsubT\mathfrak{m} \subset \mathbb{T}m⊂T be a max- imal ideal in the support of H(c)∗(SK(C),Fℓ)H(c)∗SK(C),Fâ„“H_((c))^(**)(S_(K)(C),F_(â„“))H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)∗(SK(C),Fâ„“). By work of Scholze (cf. [53, THEOREM 4.3.1]) and by the construction of Galois representations in the essentially self-dual case, we know in many cases how to associate a global modulo ℓâ„“â„“\ellâ„“ Galois representation ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm to the maximal ideal mmm\mathfrak{m}m. Therefore, the non-Eisenstein condition makes sense, and one can at least formulate Conjecture 2.2. In order to make progress on this conjecture, we impose a local representation-theoretic condition at the prime ppppp, which we treat as an auxiliary prime.
Definition 4.1. Let FFF\mathbb{F}F be a finite field of characteristic ℓâ„“â„“\ellâ„“.
(1) Let p≠ℓp≠ℓp!=â„“p \neq \ellp≠ℓ be a prime, K/QpK/QpK//Q_(p)K / \mathbb{Q}_{p}K/Qp be a finite extension, and ρ¯:Gal(K¯/K)→GLn(F)ϯ:Galâ¡(K¯/K)→GLn(F)bar(rho):Gal( bar(K)//K)rarrGL_(n)(F)\bar{\rho}: \operatorname{Gal}(\bar{K} / K) \rightarrow \mathrm{GL}_{n}(\mathbb{F})ϯ:Galâ¡(K¯/K)→GLn(F) be a continuous representation. We say that ρ¯Ï¯bar(rho)\bar{\rho}ϯ is generic if it is unramified and the eigenvalues (with multiplicity) α1,…,αn∈F¯ℓα1,…,αn∈F¯ℓalpha_(1),dots,alpha_(n)in bar(F)_(â„“)\alpha_{1}, \ldots, \alpha_{n} \in \overline{\mathbb{F}}_{\ell}α1,…,αn∈F¯ℓ of ρ¯(FrobK)ϯFrobKbar(rho)(Frob_(K))\bar{\rho}\left(\operatorname{Frob}_{K}\right)ϯ(FrobK) satisfy αi/αj≠αi/αj≠alpha_(i)//alpha_(j)!=\alpha_{i} / \alpha_{j} \neqαi/αj≠|OK/mK|OK/mK|O_(K)//m_(K)|\left|\mathcal{O}_{K} / \mathfrak{m}_{K}\right||OK/mK| for i≠ji≠ji!=ji \neq ji≠j
(2) Let FFFFF be a number field and ρ¯:Gal(F¯/F)→GLn(F)ϯ:Galâ¡(F¯/F)→GLn(F)bar(rho):Gal( bar(F)//F)rarrGL_(n)(F)\bar{\rho}: \operatorname{Gal}(\bar{F} / F) \rightarrow \mathrm{GL}_{n}(\mathbb{F})ϯ:Galâ¡(F¯/F)→GLn(F) be a continuous representation. We say that a prime p≠ℓp≠ℓp!=â„“p \neq \ellp≠ℓ is decomposed generic for ρ¯Ï¯bar(rho)\bar{\rho}ϯ if ppppp splits completely in FFFFF and, for every prime p∣pp∣pp∣p\mathfrak{p} \mid pp∣p of F,ρ¯|Gal(F¯p/Fp)F,ϯGalâ¡F¯p/FpF,( bar(rho))|_(Gal( bar(F)_(p)//F_(p)))F,\left.\bar{\rho}\right|_{\operatorname{Gal}\left(\bar{F}_{\mathfrak{p}} / F_{\mathfrak{p}}\right)}F,ϯ|Galâ¡(F¯p/Fp) is generic. We say that ρ¯Ï¯bar(rho)\bar{\rho}ϯ is decomposed generic if there exists a prime p≠ℓp≠ℓp!=â„“p \neq \ellp≠ℓ which is decomposed generic for ρ¯Ï¯bar(rho)\bar{\rho}ϯ. (If one such prime exists, then infinitely many do.)
Remark 4.2. The condition for the local representation ρ¯Ï¯bar(rho)\bar{\rho}ϯ of Gal(K¯/K)Gal(K¯/K)Gal( bar(K)//K)\mathrm{Gal}(\bar{K} / K)Gal(K¯/K) to be generic implies that any lift to characteristic 0 of ρ¯Ï¯bar(rho)\bar{\rho}ϯ corresponds under the local Langlands correspondence to a generic principal series representation of GLn(K)GLn(K)GL_(n)(K)\mathrm{GL}_{n}(K)GLn(K). Such a representation can never arise from a nonsplit inner form of GLn/KGLn/KGL_(n)//K\mathrm{GL}_{n} / KGLn/K via the Jacquet-Langlands correspondence. For this reason, a generic ρ¯Ï¯bar(rho)\bar{\rho}ϯ cannot be the modulo ℓâ„“â„“\ellâ„“ reduction of the LLLLL-parameter of a smooth representation of a nonsplit inner form of GLn/KGLn/KGL_(n)//K\mathrm{GL}_{n} / KGLn/K.
A semisimple 2-dimensional representation ρ¯Ï¯bar(rho)\bar{\rho}ϯ of Gal(Q¯/Q)Galâ¡(Q¯/Q)Gal( bar(Q)//Q)\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})Galâ¡(Q¯/Q) is either decomposed generic or it satisfies (2.1): the case where ρ¯Ï¯bar(rho)\bar{\rho}ϯ is a direct sum of two characters can be analyzed by hand, and the case where ρ¯Ï¯bar(rho)\bar{\rho}ϯ is absolutely irreducible follows from the paragraph after Theorem 3.1 in [37]. More generally, the condition for a global representation ρ¯Ï¯bar(rho)\bar{\rho}ϯ of Gal(F¯/F)Galâ¡(F¯/F)Gal( bar(F)//F)\operatorname{Gal}(\bar{F} / F)Galâ¡(F¯/F) to be decomposed generic can be ensured when ρ¯Ï¯bar(rho)\bar{\rho}ϯ has large image. For example, if ℓ>2,Fâ„“>2,Fâ„“ > 2,F\ell>2, Fâ„“>2,F is a totally real field, and ρ¯Ï¯bar(rho)\bar{\rho}ϯ is a totally odd 2-dimensional representation with nonsolvable image, then ρ¯Ï¯bar(rho)\bar{\rho}ϯ is decomposed generic (cf. [19, LEMMA 7.1.8]).
Conjecture 4.3. Let n⊂Tn⊂TnsubT\mathfrak{n} \subset \mathbb{T}n⊂T be a maximal ideal in the support of H(c)i(SK(C),Fℓ)H(c)iSK(C),Fâ„“H_((c))^(i)(S_(K)(C),F_(â„“))H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)i(SK(C),Fâ„“). Assume that ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is decomposed generic. Then the following statements hold true:
(1) if Hci(SK(C),Fℓ)m≠0HciSK(C),Fâ„“m≠0H_(c)^(i)(S_(K)(C),F_(â„“))_(m)!=0H_{c}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0Hci(SK(C),Fâ„“)m≠0, then i≤di≤di <= di \leq di≤d;
(2) if Hi(SK(C),Fℓ)m≠0HiSK(C),Fâ„“m≠0H^(i)(S_(K)(C),F_(â„“))_(m)!=0H^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0Hi(SK(C),Fâ„“)m≠0, then i≥di≥di >= di \geq di≥d.
If the Shimura varieties SKSKS_(K)S_{K}SK are compact, or if we additionally assume mmm\mathfrak{m}m to be nonEisenstein, Conjecture 4.3 implies a significant part of Conjecture 2.2 for Shimura varieties of PEL type A. Analogues of Conjecture 4.3 can be formulated (and are perhaps within reach) for other Shimura varieties, such as Siegel modular varieties.
Theorem 4.4 ([17] strengthened in [38]). Assume that GGGGG is anisotropic modulo center, so that the Shimura varieties SKSKS_(K)S_{K}SK are compact. Then Conjecture 4.3 holds true.
Theorem 4.5 ([18] strengthened in [38]). Assume that B=F,V=F2nB=F,V=F2nB=F,V=F^(2n)B=F, V=F^{2 n}B=F,V=F2n and GGGGG is a quasisplit group of unitary similitudes. Then Conjecture 4.3 holds true.
Remark 4.6. The more recent results of [38] have significantly fewer technical assumptions than the earlier ones of [17] and [18]. For example, Koshikawa's version of Theorem 4.5 allows FFFFF to be an imaginary quadratic field. It seems nontrivial to obtain this case with the methods of [18]. In the noncompact case, his results rely on the geometric constructions in [18], in particular on the semiperversity result for Shimura varieties attached to quasisplit unitary groups that is established there. As he notes, a generalization of this semiperversity result should lead to a full proof of Conjecture 4.3 for Shimura varieties of PEL type A. The more general semiperversity result will be obtained in the upcoming PhD thesis of Mafalda Santos.
In the case of Harris-Taylor Shimura varieties, Theorem 4.4 was first proved by Boyer [9]. Boyer's argument uses the integral models of Shimura varieties of Harris-Taylor type, but it is close in spirit to the argument carried out in [17] on the generic fiber. What is really interesting about Boyer's results is that he goes beyond genericity, in the following sense. Given the eigenvalues (with multiplicity) α1,…,αnα1,…,αnalpha_(1),dots,alpha_(n)\alpha_{1}, \ldots, \alpha_{n}α1,…,αn of ρ¯m(ϯmbar(rho)_(m)(:}\bar{\rho}_{\mathfrak{m}}\left(\right.ϯm( Frob p)p{:_(p))\left._{\mathfrak{p}}\right)p), with p∣pp∣pp∣p\mathfrak{p} \mid pp∣p the relevant prime of F,5F,5F,^(5)F,{ }^{5}F,5 one can define a "defect" that measures how far ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is from being generic at ppp\mathfrak{p}p. Concretely, set δp(m)δp(m)delta_(p)(m)\delta_{\mathfrak{p}}(\mathfrak{m})δp(m) to be equal to the length of the maximal chain of eigenvalues where the successive terms have ratio equal to |OFp/mFp|OFp/mFp|O_(F_(p))//m_(F_(p))|\left|\mathcal{O}_{F_{\mathfrak{p}}} / \mathfrak{m}_{F_{\mathfrak{p}}}\right||OFp/mFp|. Boyer shows that the cohomology groups H(c)i(SK(C),Fℓ)mH(c)iSK(C),Fâ„“mH_((c))^(i)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)i(SK(C),Fâ„“)m are nonzero at most in the range [d−δp(m),d+δp(m)]d−δp(m),d+δp(m)[d-delta_(p)(m),d+delta_(p)(m)]\left[d-\delta_{\mathfrak{p}}(\mathfrak{m}), d+\delta_{\mathfrak{p}}(\mathfrak{m})\right][d−δp(m),d+δp(m)]. As noted by both Emerton and Koshikawa, such a result is consistent with Arthur's conjectures on the cohomology of Shimura varieties with CCC\mathbb{C}C-coefficients and points towards a modulo ℓâ„“â„“\ellâ„“ analogue of these conjectures.
Let us also discuss the analogous vanishing result in the Hilbert case. Let FFFFF be a totally real field of degree ggggg and let G=ResF/QGL2G=ResF/Qâ¡GL2G=Res_(F//Q)GL_(2)G=\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}G=ResF/Qâ¡GL2. For a neat compact open subgroup K⊂G(Af)K⊂GAfK sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af), we let SK/QSK/QS_(K)//QS_{K} / \mathbb{Q}SK/Q be the corresponding Hilbert modular variety, of dimension ggggg.
Theorem 4.7 ([19, THEOREM A]). Let ℓ>2â„“>2â„“ > 2\ell>2â„“>2 and m⊂Tm⊂TmsubT\mathfrak{m} \subset \mathbb{T}m⊂T be a maximal ideal in the support of H(c)i(SK(C),Fℓ)H(c)iSK(C),Fâ„“H_((c))^(i)(S_(K)(C),F_(â„“))H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)i(SK(C),Fâ„“). Assume that the image of ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is not solvable, which implies that ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm is absolutely irreducible and decomposed generic. Then Hci(SK(C),Fℓ)m=Hi(SK(C),Fℓ)mHciSK(C),Fâ„“m=HiSK(C),Fâ„“mH_(c)^(i)(S_(K)(C),F_(â„“))_(m)=H^(i)(S_(K)(C),F_(â„“))_(m)H_{c}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}=H^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}Hci(SK(C),Fâ„“)m=Hi(SK(C),Fâ„“)m is nonzero only for i=gi=gi=gi=gi=g.
5 In this special case, one does not have to impose the condition that ppppp splits completely in FFFFF, and it suffices to have genericity at one prime p∣pp∣pp∣p\mathfrak{p} \mid pp∣p.
The same result holds for all quaternionic Shimura varieties, and we can even prove the analogue of Boyer's result that goes beyond genericity in all these settings. As an application, we deduce (under some technical assumptions) that the ppppp-adic local Langlands correspondence for GL2(Qp)GL2QpGL_(2)(Q_(p))\mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)GL2(Qp) occurs in the completed cohomology of Hilbert modular varieties, when ppppp is a prime that splits completely in FFFFF. This uses the axiomatic approach via patching introduced in [14] and further developed in [15,30].
We now outline the original strategy for proving Theorem 4.4, which was introduced in [17]. Let ppppp be a prime and K=KpKp⊂G(Af)K=KpKp⊂GAfK=K^(p)K_(p)sub G(A_(f))K=K^{p} K_{p} \subset G\left(\mathbb{A}_{f}\right)K=KpKp⊂G(Af) be a neat compact open subgroup. The Hodge-Tate period morphism gives rise to a TTT\mathbb{T}T-equivariant diagram
The standard comparison theorems between various cohomology theories allow us to identify H(c)∗(SK(C),Fℓ)mH(c)∗SK(C),Fâ„“mH_((c))^(**)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK(C),Fâ„“)m with H(c)∗(SK,Fℓ)mH(c)∗SK,Fâ„“mH_((c))^(**)(S_(K),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK,Fâ„“)m. The arrow on the left-hand side of (4.1) is a KpKpK_(p)K_{p}Kp-torsor, so the Hochschild-Serre spectral sequence allows us to recover H(c)∗(SK,Fℓ)mH(c)∗SK,Fâ„“mH_((c))^(**)(S_(K),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK,Fâ„“)m from H(c)∗(ςKp,Fℓ)mH(c)∗ςKp,Fâ„“mH_((c))^(**)(Ï‚_(K^(p)),F_(â„“))_(m)H_{(c)}^{*}\left(\varsigma_{K^{p}}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(Ï‚Kp,Fâ„“)m. The idea is now to compute H(c)∗(ςKp,Fℓ)mH(c)∗ςKp,Fâ„“mH_((c))^(**)(Ï‚_(K^(p)),F_(â„“))_(m)H_{(c)}^{*}\left(\varsigma_{K^{p}}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(Ï‚Kp,Fâ„“)m in two stages: first understand the complex of sheaves (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m on FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“, then compute the total cohomology using the Leray-Serre spectral sequence.
Two miraculous things happen that greatly simplify the structure of (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m. The first is that (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m behaves like a perverse sheaf on FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“. This is because πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT is simultaneously affinoid, as discussed after Theorem 3.1, and partially proper, because the Shimura varieties were assumed to be compact. In particular, the restriction of (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m to a highest-dimensional stratum in its support is concentrated in one degree. By the computation of the fibers of πHTÏ€HTpi_(HT)\pi_{\mathrm{HT}}Ï€HT, this implies that the cohomology groups RΓ(ℑgb,Zℓ)mRΓℑgb,Zâ„“mR Gamma((â„‘g)^(b),Z_(â„“))_(m)R \Gamma\left(\mathfrak{\Im g}{ }^{b}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}}RΓ(â„‘gb,Zâ„“)m are concentrated in one degree and torsion-free. The second miracle is that, whenever the group Gb(Qp)GbQpG_(b)(Q_(p))G_{b}\left(\mathbb{Q}_{p}\right)Gb(Qp) acting on sgbsgbsg^(b)\mathfrak{s g}^{b}sgb comes from a nonquasisplit inner form, the localization RΓ(sgb,Qℓ)mRΓsgb,Qâ„“mR Gamma(sg^(b),Q_(â„“))_(m)R \Gamma\left(\mathfrak{s g}^{b}, \mathbb{Q}_{\ell}\right)_{\mathfrak{m}}RΓ(sgb,Qâ„“)m vanishes. This uses the genericity of ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm at each p∣pp∣pp∣p\mathfrak{p} \mid pp∣p and suggests that the cohomology of Igusa varieties satisfies some form of local-global compatibility. Finally, the condition that ppppp splits completely in FFFFF guarantees that the only Newton stratum for which GbGbG_(b)G_{b}Gb is quasisplit is the ordinary one. Therefore, the hypotheses of Theorem 4.4 guarantee that (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m is as simple as possible - it is supported in one degree on a zero-dimensional stratum!
The computation of RΓ(Igb,Qℓ)mRΓIgb,Qâ„“mR Gamma(Ig^(b),Q_(â„“))_(m)R \Gamma\left(\mathrm{Ig}^{b}, \mathbb{Q}_{\ell}\right)_{\mathfrak{m}}RΓ(Igb,Qâ„“)m, at least at the level of the Grothendieck group, can be done using the trace formula method pioneered by Shin [60]. This is the method used for Shimura varieties of PEL type A in [17] and [18]. For inner forms of ResF/QGL2ResF/Qâ¡GL2Res_(F//Q)GL_(2)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}ResF/Qâ¡GL2, with FFFFF a totally real field, one can avoid these difficult computations, cf. [19]. In this setting, one can reinterpret results of Tian-Xiao [62] on geometric instances of the Jacquet-Langlands correspondence as giving rise to exotic isomorphisms between Igusa varieties arising from different Shimura varieties. This is what happens for the basic stratum in Example 3.4. Then one can conclude by applying the classical Jacquet-Langlands correspondence.
In [38], Koshikawa introduces a novel and complementary strategy for proving these kinds of vanishing theorems. He shows that, under the same genericity assumption in Definition 4.1, only the restriction of (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m to the ordinary locus contributes to the total cohomology of the Shimura variety. To achieve this, he proves the analogous generic vanishing theorem for the cohomology RΓc(Mb,Zℓ)mpRΓcMb,Zâ„“mpRGamma_(c)(M^(b),Z_(â„“))_(m_(p))R \Gamma_{c}\left(\mathcal{M}^{b}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}_{p}}RΓc(Mb,Zâ„“)mp of the Rapoport-Zink space, where mpmpm_(p)\mathfrak{m}_{p}mp is a maximal ideal of the local spherical Hecke algebra at ppppp. This relies on the recent work of Fargues-Scholze on the geometrization of the local Langlands conjecture [28].
Koshikawa's strategy is more flexible, allowing him to handle with ease the case where FFFFF is an imaginary quadratic field. On the other hand, the original approach also gives information about the complexes of sheaves (RπHT∗Fℓ)mRÏ€HT∗Fâ„“m(Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m, rather than just about the cohomology groups H(c)∗(SK(C),Fℓ)mH(c)∗SK(C),Fâ„“mH_((c))^(**)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK(C),Fâ„“)m. These complexes should play an important role for questions of local-global compatibility in Fargues's geometrization conjecture, cf. [26, 87].
5. COHOMOLOGY WITH MOD ppppp AND ppppp-ADIC COEFFICIENTS
The most general method for constructing ppppp-adic families of automorphic forms from the cohomology of locally symmetric spaces is via completed cohomology. First introduced by Emerton in [24], this has the following definition:
where Kp⊂G(Af)Kp⊂GAfK^(p)sub G(A_(f))K^{p} \subset G\left(\mathbb{A}_{f}\right)Kp⊂G(Af) is a sufficiently small, fixed tame level, and Kp⊂G(Qp)Kp⊂GQpK_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right)Kp⊂G(Qp) runs over compact open subgroups. This space has an action of the spherical Hecke algebra TTT\mathbb{T}T, built from Hecke operators away from ppppp, as well as an action of the group G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). One can make the analogous definition for completed cohomology with compact support, and a variant gives completed homology and completed Borel-Moore homology. See [25] for an excellent survey that gives motivation, examples, and sketches the basic properties of these spaces.
Motivated by heuristics from the ppppp-adic Langlands programme, Calegari and Emerton made several conjectures about the range of degrees in which one can have nonzero completed (co)homology and about the codimension of completed homology over the completed group rings Zp[[Kp]]Zp[[Kp]]Z_(p)[[K_(p)]]\mathbb{Z}_{p} \llbracket K_{p} \rrbracketZp[[Kp]]. See [11, coNJECTURE 1.5] for the original formulation and [32, CONJECTURE 1.3] for a slightly different formulation, which emphasizes the natural implications between the various conjectures. In particular, Calegari-Emerton conjectured that
H~ci(Kp,Zp)=H~i(Kp,Zp)=0 for i>q0H~ciKp,Zp=H~iKp,Zp=0 for i>q0tilde(H)_(c)^(i)(K^(p),Z_(p))= tilde(H)^(i)(K^(p),Z_(p))=0quad" for "i > q_(0)\tilde{H}_{c}^{i}\left(K^{p}, \mathbb{Z}_{p}\right)=\tilde{H}^{i}\left(K^{p}, \mathbb{Z}_{p}\right)=0 \quad \text { for } i>q_{0}H~ci(Kp,Zp)=H~i(Kp,Zp)=0 for i>q0
For Shimura varieties of preabelian type, the Calegari-Emerton conjectures were proved by Hansen-Johansson in [32], building on work of Scholze who proved the vanishing of completed cohomology with compact support for Shimura varieties of Hodge type [53].
We sketch Scholze's argument, which illustrates the role of ppppp-adic geometry in this result. It is enough to show that
In [20] and [16], we study Shimura varieties with unipotent level at ppppp. More precisely, assume that (G,X)(G,X)(G,X)(G, X)(G,X) is a Shimura datum of Hodge type and that GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split. Choose a split model of GGGGG and a Borel subgroup BBBBB over ZpZpZ_(p)\mathbb{Z}_{p}Zp, and let U⊂BU⊂BU sub BU \subset BU⊂B be the unipotent radical.
Theorem 5.1 ([16, THEOREM 1.1]). Let H⊆U(Zp)H⊆UZpH sube U(Z_(p))H \subseteq U\left(\mathbb{Z}_{p}\right)H⊆U(Zp) be a closed subgroup. We have
limKp⊇HHci(SKpKp(C),Fp)=0 for i>dlimKp⊇H HciSKpKp(C),Fp=0 for i>dlim_(K_(p)supe H)H_(c)^(i)(S_(K^(p)K_(p))(C),F_(p))=0quad" for "i > d\underset{K_{p} \supseteq H}{\lim } H_{c}^{i}\left(S_{K^{p} K_{p}}(\mathbb{C}), \mathbb{F}_{p}\right)=0 \quad \text { for } i>dlimKp⊇HHci(SKpKp(C),Fp)=0 for i>d
This result is stronger than the Calegari-Emerton conjecture for completed cohomology with compact support, since we can take H={1}H={1}H={1}H=\{1\}H={1} and recover Scholze's result discussed above. In addition to the argument sketched above, the key new idea needed for Theorem 5.1 is that the Bruhat decomposition on the Hodge-Tate period domain FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“ remembers how far different subspaces of SKpU(Zp)∗SKpUZp∗S_(K^(p)U(Z_(p)))^(**)\mathcal{S}_{K^{p} U\left(\mathbb{Z}_{p}\right)}^{*}SKpU(Zp)∗ are from being perfectoid.
Example 5.2. Assume that G=GL2/QG=GL2/QG=GL_(2)//QG=\mathrm{GL}_{2} / \mathbb{Q}G=GL2/Q, so that we are working in the modular curve case. The Bruhat decomposition is given by P1, ad =A1, ad ⊔{∞}P1, ad =A1, ad ⊔{∞}P^(1," ad ")=A^(1," ad ")⊔{oo}\mathbb{P}^{1, \text { ad }}=\mathbb{A}^{1, \text { ad }} \sqcup\{\infty\}P1, ad =A1, ad ⊔{∞}, with the two Bruhat cells in natural bijection with the two components of the ordinary locus in (3.5). We have a morphism of sites
where we take the quotient |P1,ad|/U(Zp)P1,ad/UZp|P^(1,ad)|//U(Z_(p))\left|\mathbb{P}^{1, \mathrm{ad}}\right| / U\left(\mathbb{Z}_{p}\right)|P1,ad|/U(Zp) only as a topological space. The preimage of |A1, ad |/U(Zp)A1, ad /UZp|A^(1," ad ")|//U(Z_(p))\left|\mathbb{A}^{1, \text { ad }}\right| / U\left(\mathbb{Z}_{p}\right)|A1, ad |/U(Zp) in SKpU(Zp)SKpUZpS_(K^(p)U(Z_(p)))S_{K^{p} U\left(\mathbb{Z}_{p}\right)}SKpU(Zp) is a perfectoid space, as proved by Ludwig in [42]. The preimage of |∞|/U(Zp)|∞|/UZp|oo|//U(Z_(p))|\infty| / U\left(\mathbb{Z}_{p}\right)|∞|/U(Zp) has a ZpZpZ_(p)\mathbb{Z}_{p}Zp-cover that is an affinoid perfectoid space. This allows us to bound the support of each RiπHT∗/U(Zp)(L+/p)RiÏ€HT∗/UZpL+/pR^(i)pi_(HT**//U(Z_(p)))(L^(+)//p)R^{i} \pi_{\mathrm{HT} * / U\left(\mathbb{Z}_{p}\right)}\left(\mathscr{L}^{+} / p\right)RiÏ€HT∗/U(Zp)(L+/p), and we conclude by the Leray spectral sequence.
More generally, the Bruhat decomposition G=⨆w∈WPμBwPμG=⨆w∈WPμ BwPμG=⨆_(w inW^(P mu))BwP_(mu)G=\bigsqcup_{w \in W^{P \mu}} B w P_{\mu}G=⨆w∈WPμBwPμ gives a decomposition Fℓ=⨆w∈WPμFℓwFâ„“=⨆w∈WPμ Fâ„“wFâ„“=⨆_(w inW^(P_(mu)))Fâ„“^(w)\mathscr{F} \ell=\bigsqcup_{w \in W^{P_{\mu}}} \mathscr{F} \ell^{w}Fâ„“=⨆w∈WPμFâ„“w into locally closed Schubert cells indexed by certain Weyl group elements. For each Fℓw/U(Zp)Fâ„“w/UZpFâ„“^(w)//U(Z_(p))\mathscr{F} \ell^{w} / U\left(\mathbb{Z}_{p}\right)Fâ„“w/U(Zp), we can quantify how far its preimage in SKpU(Zp)∗SKpUZp∗S_(K^(p)U(Z_(p)))^(**)S_{K^{p} U\left(\mathbb{Z}_{p}\right)}^{*}SKpU(Zp)∗ is from being a perfectoid space, which depends on the length of the Weyl group element wwwww. The assumption that GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split guarantees that all the Weyl group elements lie in the ordinary locus inside FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“, which greatly simplifies the analysis. However, the analogue of Theorem 5.1 may hold even without the assumption that GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split, and even when the ordinary locus is empty. There is some evidence in this direction, e.g., by using embeddings into higher-dimensional Shimura varieties attached to split groups, or by using the results of [36] to handle the Harris-Taylor case, as in the upcoming PhD thesis of Louis Jaburi.
The Bruhat decomposition on FℓFâ„“Fâ„“\mathscr{F} \ellFâ„“ has more recently been used by Boxer and Pilloni to define a version of higher Coleman theory indexed by each w∈WPμw∈WPμw inW^(P_(mu))w \in W^{P_{\mu}}w∈WPμ in [8]. The development of higher Coleman and higher Hida theories shows that the geometric theory of ppppp-adic automorphic forms on Shimura varieties is much richer than previously expected. Furthermore, the Bruhat decomposition indicates the form a ppppp-adic Eichler-Shimura isomorphism should take, relating completed cohomology to these more geometric theories. In joint work in progress with Mantovan and Newton, we use the geometry described in Example 5.2 to give a new proof of the ordinary Eichler-Shimura isomorphism due to Ohta [46, 47]. Our result decomposes the ordinary completed cohomology of the modular curve in terms of Hida theory and higher Hida theory, the latter recently developed by Boxer and Pilloni in [7].
While the focus of this article has been the cohomology of Shimura varieties, Theorems 4.5 and 5.1 have surprising applications to understanding the cohomology of more general locally symmetric spaces. For example, let FFFFF be an imaginary CMCMCM\mathrm{CM}CM field and G=G=G=G=G=ResF/QGLnResF/Qâ¡GLnRes_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Qâ¡GLn. Then GGGGG can be realized as the Levi quotient of the Siegel maximal parabolic of a quasisplit unitary group G~G~tilde(G)\tilde{G}G~. The Borel-Serre compactification X~K~BSX~K~BStilde(X)_( tilde(K))^(BS)\tilde{X}_{\tilde{K}}^{\mathrm{BS}}X~K~BS for the locally symmetric spaces associated with the unitary group G~G~tilde(G)\tilde{G}G~ gives rise to a Hecke-equivariant long exact sequence of the form
where ∂X~K~=X~K~BS∖X~K~∂X~K~=X~K~BS∖X~K~del tilde(X)_( tilde(K))= tilde(X)_( tilde(K))^(BS)\\ tilde(X)_( tilde(K))\partial \tilde{X}_{\tilde{K}}=\tilde{X}_{\tilde{K}}^{\mathrm{BS}} \backslash \tilde{X}_{\tilde{K}}∂X~K~=X~K~BS∖X~K~ is the boundary of the Borel-Serre compactification. The usual and compactly supported cohomology of X~K~X~K~tilde(X)_( tilde(K))\tilde{X}_{\tilde{K}}X~K~ can be simplified to some extent by applying either of the two vanishing theorems. On the other hand, the cohomology of XKXKX_(K)X_{K}XK can be shown to contribute to the cohomology of ∂X~K~∂X~K~del tilde(X)_( tilde(K))\partial \tilde{X}_{\tilde{K}}∂X~K~, in some more or less controlled fashion.
Let m⊂Tm⊂TmsubT\mathfrak{m} \subset \mathbb{T}m⊂T be a non-Eisenstein maximal ideal in the support of RΓ(XK,Zℓ)RΓXK,Zâ„“R Gamma(X_(K),Z_(â„“))R \Gamma\left(X_{K}, \mathbb{Z}_{\ell}\right)RΓ(XK,Zâ„“) and let T(K)mT(K)mT(K)_(m)\mathbb{T}(K)_{\mathfrak{m}}T(K)m denote the quotient of TTT\mathbb{T}T that acts faithfully on RΓ(XK,Zℓ)mRΓXK,Zâ„“mR Gamma(X_(K),Z_(â„“))_(m)R \Gamma\left(X_{K}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}}RΓ(XK,Zâ„“)m. In addition to the residual Galois representation ρ¯mϯmbar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ϯm, Scholze associates to mmm\mathfrak{m}m a deformation ρmÏmrho_(m)\rho_{\mathfrak{m}}Ïm valued in T(K)m/IT(K)m/IT(K)_(m)//I\mathbb{T}(K)_{\mathfrak{m}} / IT(K)m/I, for some nilpotent ideal IIIII. This was subsequently shown by Newton and Thorne in [45] to satisfy I4=0I4=0I^(4)=0I^{4}=0I4=0. In [20], we used a variant of Theorem 5.1 together with the excision sequence (6.1) to eliminate this nilpotent ideal entirely, under the assumption that ℓâ„“â„“\ellâ„“ splits
completely in the CM field FFFFF. This leads to a more natural statement on the existence of Galois representations in this setting.
The Galois representations ρmÏmrho_(m)\rho_{\mathfrak{m}}Ïm are expected to satisfy a certain property known as local-global compatibility, which is particularly subtle to state and prove at primes above ℓâ„“â„“\ellâ„“. For example, after inverting ℓâ„“â„“\ellâ„“, the ρmÏmrho_(m)\rho_{\mathfrak{m}}Ïm are expected to be de Rham, in the sense of Fontaine, but it is less clear what the right condition should be for torsion Galois representations. In another application, Theorem 4.5 is crucially used in [1] together with the excision sequence (6.1) to prove that ρmÏmrho_(m)\rho_{\mathfrak{m}}Ïm satisfies the expected local-global compatibility at primes above ℓâ„“â„“\ellâ„“ in two restricted families of cases: the ordinary case and the FontaineLaffaille case. 66^(6){ }^{6}6 In joint work in progress with Newton, we should be able to extend these methods to cover significantly more.
The local-global compatibility results established in [1] are already extremely useful: they help us implement the Calegari-Geraghty method unconditionally for the first time in arbitrary dimension. A striking application is the following result.
Theorem 6.1 ([1, THEOREM 1.0.1]). Let FFFFF be a CM field and E/FE/FE//FE / FE/F be an elliptic curve that does not have complex multiplication. Then EEEEE is potentially automorphic and satisfies the Sato-Tate conjecture.
The potential automorphy of EEEEE was established at the same time in work of BoxerCalegari-Gee-Pilloni [6], who also showed the potential automorphy of abelian surfaces over totally real fields. Their work relies on the Calegari-Geraghty method for the coherent cohomology of Shimura varieties and uses a preliminary version of higher Hida theory, due to Pilloni, as a key ingredient.
ACKNOWLEDGMENTS
I am grateful to my many colleagues and collaborators who have discussed mathematics in general and Shimura varieties in particular with me - I especially want to thank Frank Calegari, Matthew Emerton, Toby Gee, Dan Gulotta, Pol van Hoften, Christian Johansson, Elena Mantovan, Sophie Morel, James Newton, Vytas Paškūnas, Vincent Pilloni, Peter Scholze, Sug Woo Shin, Matteo Tamiozzo, and Richard Taylor. I am grateful to Toby Gee, Louis Jaburi, Teruhisa Koshikawa, Peter Scholze, and Matteo Tamiozzo for their comments on an earlier draft of this article.
FUNDING
This work was partially supported by a Royal Society University Research Fellowship, by a Leverhulme Prize, and by ERC Starting Grant No. 804176.
6 Up to possibly enlarging the nilpotent ideal IIIII. It is not clear how to remove the nilpotent ideal from the statement of local-global compatibility at ℓ=pâ„“=pâ„“=p\ell=pâ„“=p.
REFERENCES
[1] P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne, Potential automorphy over CM fields. 2018, arXiv:1812.09999.
[2] J. Anschütz, Reductive group schemes over the Fargues-Fontaine curve. Math. Ann. 374 (2019), no. 3-4, 1219-1260.
[3] J. Arthur, L2L2L^(2)L^{2}L2-cohomology and automorphic representations. In In Canadian Mathematical Society. 1945-1995, Vol. 3, pp. 1-17, Canadian Math. Soc., Ottawa, ON, 1996.
[4] A. Ash, Galois representations attached to mod ppppp cohomology of G L(n,Z)L(n,Z)L(n,Z)L(n, Z)L(n,Z). Duke Math. J. 65 (1992), no. 2, 235-255.
[5] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edn., Math. Surveys Monogr. 67, American Mathematical Society, Providence, RI, 2000.
[6] G. Boxer, F. Calegari, T. Gee, and V. Pilloni, Abelian Surfaces over totally real fields are Potentially Modular. 2018, arXiv:1812.09269.
[7] G. Boxer and V. Pilloni, Higher Hida and Coleman theories on the modular curve. 2020, arXiv:2002.06845.
[8] G. Boxer and V. Pilloni, Higher Coleman theory. 2021, arXiv:2110.10251.
[10] F. Calegari, Reciprocity in the Langlands program since Fermat's Last Theorem. 2021, arXiv:2109.14145.
[11] F. Calegari and M. Emerton, Completed cohomology-a survey. In Non-abelian fundamental groups and Iwasawa theory, pp. 239-257, London Math. Soc. Lecture Note Ser. 393, Cambridge Univ. Press, Cambridge, 2012.
[12] F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor-Wiles method. Invent. Math. 211 (2018), no. 1, 297-433.
[13] A. Caraiani, Perfectoid Shimura varieties. In Perfectoid spaces: Lectures from the 2017 Arizona Winter School, pp. 245-297, American Mathematical Society, Providence, RI, 2019.
[14] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, and S. W. Shin, Patching and the ppppp-adic local Langlands correspondence. Camb. J. Math. 4 (2016), no. 2, 197-287.
[15] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, and S. W. Shin, Patching and the ppppp-adic Langlands program for GL2(Qp)GL2QpGL_(2)(Q_(p))\mathrm{G} L_{2}\left(\mathbb{Q}_{p}\right)GL2(Qp). Compos. Math. 154 (2018), no. 3, 503-548.
[16] A. Caraiani, D. R. Gulotta, and C. Johansson, Vanishing theorems for Shimura varieties at unipotent level. Journal of the EMS (to appear), arXiv:1910.09214.
[17] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2) 186 (2017), no. 3, 649-766.
[18] A. Caraiani and P. Scholze, On the generic part of the cohomology of noncompact unitary Shimura varieties. 2019, arXiv:1909.01898.
[20] A. Caraiani, D. R. Gulotta, C.-Y. Hsu, C. Johansson, L. Mocz, E. Reinecke, and S.-C. Shih, Shimura varieties at level Γ1(p∞)Γ1p∞Gamma_(1)(p^(oo))\Gamma_{1}\left(p^{\infty}\right)Γ1(p∞) and Galois representations. Compos. Math. 156 (2020), no. 6, 1152-1230.
[21] G. Chenevier and M. Harris, Construction of automorphic Galois representations, II. Camb. J. Math. 1 (2013), no. 1, 53-73.
[24] M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Invent. Math. 164 (2006), no. 1, 1-84.
[25] M. Emerton, Completed cohomology and the ppppp-adic Langlands program. In Proceedings of the International Congress of Mathematicians, 2014. Volume II, pp. 319-342, Kyung Moon Sa, Seoul, 2014.
[26] L. Fargues, Geometrization of the local Langlands correspondence: an overview. 2016, arXiv:1602.00999.
[30] T. Gee and J. Newton, Patching and the completed homology of locally symmetric spaces. J. Inst. Math. Jussieu (2016).
[31] P. Hamacher, The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type. J. Algebraic Geom. 28 (2019), no. 4, 721-749.
[32] D. Hansen and C. Johansson, Perfectoid Shimura varieties and the CalegariEmerton conjectures. 2020, arXiv:2011.03951.
[33] M. Harris, K.-W. Lan, R. Taylor, and J. Thorne, On the rigid cohomology of certain Shimura varieties. Res. Math. Sci. 3 (2016), no. 3, 37.
[34] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. Ann. of Math. Stud. 151, Princeton University Press, Princeton, NJ, 2001.
[35] S. Howe, The spectral ppppp-adic Jacquet-Langlands correspondence and a question of Serre. Compos. Math. (to appear).
[36] C. Johansson, J. Ludwig, and D. Hansen, A quotient of the Lubin-Tate tower II. Math. Ann. 380 (2021), no. 1-2, 43-89.
[37] T. Koshikawa, Vanishing theorems for the mod ppppp cohomology of some simple Shimura varieties. Forum Math. Sigma 8 (2020), Paper No. e38.
[38] T. Koshikawa, On the generic part of the cohomology of local and global Shimura varieties. 2021, arXiv:2106.10602.
[39] K.-W. Lan, An example-based introduction to Shimura varieties. 2018.
[40] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties. Duke Math. J. 161 (2012), no. 6, 1113−11701113−11701113-11701113-11701113−1170.
[41] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties. Adv. Math. 242 (2013), 228-286.
[42] J. Ludwig, A quotient of the Lubin-Tate tower. Forum Math. Sigma 5 (2017), e17.
[43] E. Mantovan, On the cohomology of certain PEL-type Shimura varieties. Duke Math. J. 129 (2005), no. 3, 573-610.
[45] J. Newton and J. A. Thorne, Torsion Galois representations over CM fields and Hecke algebras in the derived category. Forum Math. Sigma 4 (2016), e21, 88.
[46] M. Ohta, On the ppppp-adic Eichler-Shimura isomorphism for ΛΛLambda\LambdaΛ-adic cusp forms. JJJJJ. Reine Angew. Math. 463 (1995), 49-98.
[52] P. Scholze, ppppp-adic Hodge theory for rigid-analytic varieties. Forum Math. Pi 1 (2013), e1, 77.
[53] P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2) 182 (2015), no. 3, 945-1066.
[54] P. Scholze, Étale cohomology of diamonds. 2017, arXiv:1709.07343.
[55] P. Scholze and J. Weinstein, Moduli of ppppp-divisible groups. Camb. J. Math. 1 (2013), no. 2, 145-237.
[56] P. Scholze and J. Weinstein, Berkeley Lectures on p-adic Geometry. Ann. of Math. Stud. 207, Princeton University Press, Princeton, NJ, 2020, 264 pp.
[57] J. Sempliner, On the almost-product structure on the moduli stacks of parahoric global G-shtuka. PhD thesis, Princeton University, 2021.
[59] S. W. Shin, Counting points on Igusa varieties. Duke Math. J. 146 (2009), no. 3, 509-568.
[60] S. W. Shin, A stable trace formula for Igusa varieties. J. Inst. Math. Jussieu 9 (2010), no. 4, 847-895.
[61] S. W. Shin, Galois representations arising from some compact Shimura varieties. Ann. of Math. (2) 173 (2011), no. 3, 1645-1741.
[62] Y. Tian and L. Xiao, On Goren-Oort stratification for quaternionic Shimura varieties. Compos. Math. 152 (2016), no. 10, 2134-2220.
[63] E. Viehmann, On Newton strata in the BdR+BdR+B_(dR)^(+)B_{\mathrm{dR}}^{+}BdR+-Grassmannian. 2021, arXiv:2101.07510.
[64] L. Xiao and X. Zhu, Cycles on Shimura varieties via geometric Satake. 2017, arXiv:1707.05700.
ANA CARAIANI
180 Queen's Gate, London SW7 2AZ, United Kingdom, a.caraiani @ imperial.ac.uk
ON THE BRUMER-STARK CONJECTURE AND REFINEMENTS
SAMIT DASGUPTA AND MAHESH KAKDE
Abstract
We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at s=1s=1s=1s=1s=1 (equivalently, s=0s=0s=0s=0s=0 ). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form.
The conjecture considers a totally real field FFFFF and a finite abelian CM extension H/FH/FH//FH / FH/F. It states the existence of ppppp-units in HHHHH whose valuations at places above ppppp are related to the special values of the LLLLL-functions of the extension H/FH/FH//FH / FH/F at s=0s=0s=0s=0s=0. Essentially equivalently, the conjecture states that a Stickelberger element associated to H/FH/FH//FH / FH/F annihilates the (appropriately smoothed) class group of HHHHH.
We describe our recent proofs of the Brumer-Stark conjecture away from 2. The conjecture has been refined by many authors in multiple directions. We state some of these refinements and our results towards them. The key technique involved in the proofs is Ribet's method.
One of the refinements we discuss is an exact ppppp-adic analytic formula for Brumer-Stark units stated by the first author and his collaborators. We describe this formula and highlight some salient points of its proof. Since the Brumer-Stark units along with other easily described elements generate the maximal abelian extension of a totally real field, our results can be viewed as an explicit class theory for such fields. This can be considered a ppppp-adic version of Hilbert's 12th problem.
MATHEMATICS SUBJECT CLASSIFICATION 2020
Primary 11R37; Secondary 11R42
KEYWORDS
Stark conjectures, Brumer-Stark units, Explicit class field theory
1. BACKGROUND AND MOTIVATION
Dirichlet's class number formula, conjectured for quadratic fields by Jacobi in 1832 and proven by Dirichlet in 1839, is one of the earliest examples of a relationship between leading terms of LLLLL-functions and global arithmetic invariants. Let FFFFF be a number field with ring of integers OFOFO_(F)O_{F}OF. The Dedekind zeta function associated with FFFFF is defined as
where a runs through the nonzero ideals in OFOFO_(F)O_{F}OF. The function ζF(s)ζF(s)zeta_(F)(s)\zeta_{F}(s)ζF(s) generalizes Riemann's zeta function and has a meromorphic continuation to the whole complex plane with only a simple pole at s=1s=1s=1s=1s=1. Dirichlet's class number formula, which is proved using a "geometry of numbers" approach, evaluates the residue at s=1s=1s=1s=1s=1 :
Here r1r1r_(1)r_{1}r1 is the number of real embeddings of FFFFF and 2r22r22r_(2)2 r_{2}2r2 is the number of complex embeddings of FFFFF. Further, hFhFh_(F)h_{F}hF and RFRFR_(F)R_{F}RF denote the class number and regulator (defined below) of FFFFF, respectively, while wFwFw_(F)w_{F}wF denotes the number of roots of unity in FFFFF and DFDFD_(F)D_{F}DF is the discriminant of F/QF/QF//QF / \mathbf{Q}F/Q. The meromorphic function ζF(s)ζF(s)zeta_(F)(s)\zeta_{F}(s)ζF(s) satisfies a functional equation relating ζF(s)ζF(s)zeta_(F)(s)\zeta_{F}(s)ζF(s) and ζF(1−s)ζF(1−s)zeta_(F)(1-s)\zeta_{F}(1-s)ζF(1−s). Using this functional equation, Dirichlet's class number formula can be restated as giving the leading term of the Taylor expansion of ζF(s)ζF(s)zeta_(F)(s)\zeta_{F}(s)ζF(s) at s=0s=0s=0s=0s=0 :
Artin described a theory of LLLLL-functions generalizing the Dedekind zeta function. Let GFGFG_(F)G_{F}GF be the absolute Galois group of FFFFF. A Dirichlet character for FFFFF (or a degree 1 Artin character of FFFFF ) is a homomorphism χ:GF→C×χ:GF→C×chi:G_(F)rarrC^(xx)\chi: G_{F} \rightarrow \mathbf{C}^{\times}χ:GF→C×with finite image. Class field theory identifies χχchi\chiχ with a function, again denoted by χχchi\chiχ, from the set of nonzero ideals of OFOFO_(F)O_{F}OF to CCC\mathbf{C}C. Define
Again L(χ,s)L(χ,s)L(chi,s)L(\chi, s)L(χ,s) has a meromorphic continuation to the whole complex plane with only a simple pole at s=1s=1s=1s=1s=1 if χχchi\chiχ is trivial. If H/FH/FH//FH / FH/F is a Galois extension with finite abelian Galois group G=Gal(H/F)G=Galâ¡(H/F)G=Gal(H//F)G=\operatorname{Gal}(H / F)G=Galâ¡(H/F), then we can view any character χ∈G^=Hom(G,C∗)χ∈G^=Homâ¡G,C∗chi in hat(G)=Hom(G,C^(**))\chi \in \hat{G}=\operatorname{Hom}\left(G, \mathbf{C}^{*}\right)χ∈G^=Homâ¡(G,C∗) as a Dirichlet character for FFFFF, and we have the Artin factorization formula
(1.2)ζH(s)=∏χ∈G^L(χ,s)(1.2)ζH(s)=âˆÏ‡âˆˆG^ L(χ,s){:(1.2)zeta_(H)(s)=prod_(chi in hat(G))L(chi","s):}\begin{equation*}
\zeta_{H}(s)=\prod_{\chi \in \hat{G}} L(\chi, s) \tag{1.2}
\end{equation*}(1.2)ζH(s)=âˆÏ‡âˆˆG^L(χ,s)
Dirichlet's class number formula (1.1) for the field HHHHH gives the leading term of the left-hand side of (1.2) at s=0s=0s=0s=0s=0. Stark asked for an analogous formula for L(χ,s)L(χ,s)L(chi,s)L(\chi, s)L(χ,s) at s=0s=0s=0s=0s=0 for each character χχchi\chiχ, thereby giving a canonical factorization of the term hHRH/wHhHRH/wHh_(H)R_(H)//w_(H)h_{H} R_{H} / w_{H}hHRH/wH. This led to the formulation of the abelian Stark conjecture, which we state in Section 2. This statement involves the choice of places of FFFFF that split completely in HHHHH. After stating Stark's conjecture, we restrict in the remainder of the paper to the case that the splitting places of FFFFF are finite.
Since the associated LLLLL-values here are algebraic, one can make progress on the conjectures through ppppp-adic techniques such as ppppp-adic Galois cohomology. To obtain nonzero LLLLL-values (and hence have nontrivial statements), parity conditions force us to restrict to the setting that FFFFF is a totally real field and HHHHH is a CM field.
Stark's conjecture at finite places has a natural restatement in terms of annihilators of class groups as formulated in the Brumer-Stark conjecture. We recall the statement and its refinements in Sections 4-5. The rest of the paper is taken up in describing the statement and proofs of our results toward the Brumer-Stark conjecture and its refinements.
2. STARK'S CONJECTURE
Let us first reformulate Dirichlet's class number formula.
For any place wwwww of FFFFF we normalize the absolute value |⋅|w:Fw∗→R|â‹…|w:Fw∗→R|*|_(w):F_(w)^(**)rarrR|\cdot|_{w}: F_{w}^{*} \rightarrow \mathbf{R}|â‹…|w:Fw∗→R by
|u|w={|u| if w is real |u|2 if w is complex Nw−ordw(u) if w is nonarchimedean. |u|w=|u| if w is real |u|2 if w is complex Nw−ordwâ¡(u) if w is nonarchimedean. |u|_(w)={[|u|," if "w" is real "],[|u|^(2)," if "w" is complex "],[Nw^(-ord_(w)(u))," if "w" is nonarchimedean. "]:}|u|_{w}= \begin{cases}|u| & \text { if } w \text { is real } \\ |u|^{2} & \text { if } w \text { is complex } \\ \mathrm{N} w^{-\operatorname{ord}_{w}(u)} & \text { if } w \text { is nonarchimedean. }\end{cases}|u|w={|u| if w is real |u|2 if w is complex Nw−ordwâ¡(u) if w is nonarchimedean.Â
For a finite set of places SSSSS of FFFFF, let XSXSX_(S)X_{S}XS denote the degree zero subgroup of the free abelian group on SSSSS. Let u1,…,ur1+r2−1u1,…,ur1+r2−1u_(1),dots,u_(r_(1)+r_(2)-1)u_{1}, \ldots, u_{r_{1}+r_{2}-1}u1,…,ur1+r2−1 be a set of generators of the free abelian group OF∗/μFOF∗/μFO_(F)^(**)//mu_(F)O_{F}^{*} / \mu_{F}OF∗/μF. Let S∞S∞S_(oo)S_{\infty}S∞ be the set of archimedean places of FFFFF.
induces an isomorphism ROF∗→RXS∞ROF∗→RXS∞RO_(F)^(**)rarrRX_(S_(oo))\mathbf{R} O_{F}^{*} \rightarrow \mathbf{R} X_{S_{\infty}}ROF∗→RXS∞. Here and throughout, RXS∞RXS∞RX_(S_(oo))\mathbf{R} X_{S_{\infty}}RXS∞ denotes R⊗ZXS∞R⊗ZXS∞Rox_(Z)X_(S_(oo))\mathbf{R} \otimes_{\mathbf{Z}} X_{S_{\infty}}R⊗ZXS∞, etc. Let w1,…,wr1+r2w1,…,wr1+r2w_(1),dots,w_(r_(1)+r_(2))w_{1}, \ldots, w_{r_{1}+r_{2}}w1,…,wr1+r2 denote the archimedean places of FFFFF. Then
(2.1){wi−w1:2≤i≤r+1},r=r1+r2−1(2.1)wi−w1:2≤i≤r+1,r=r1+r2−1{:(2.1){w_(i)-w_(1):2 <= i <= r+1}","quad r=r_(1)+r_(2)-1:}\begin{equation*}
\left\{w_{i}-w_{1}: 2 \leq i \leq r+1\right\}, \quad r=r_{1}+r_{2}-1 \tag{2.1}
\end{equation*}(2.1){wi−w1:2≤i≤r+1},r=r1+r2−1
is an integral basis of XS∞XS∞X_(S_(oo))X_{S_{\infty}}XS∞. Let RFRFR_(F)R_{F}RF be the absolute value of the determinant of the isomorphism between ROF∗ROF∗RO_(F)^(**)\mathbf{R} O_{F}^{*}ROF∗ and RXS∞RXS∞RX_(S_(oo))\mathbf{R} X_{S_{\infty}}RXS∞ with respect to the bases {u1,…,ur1+r2−1}u1,…,ur1+r2−1{u_(1),dots,u_(r_(1)+r_(2)-1)}\left\{u_{1}, \ldots, u_{r_{1}+r_{2}-1}\right\}{u1,…,ur1+r2−1} and (2.1), respectively. Up to a sign, Dirichlet's class number formula can be restated as follows:
(i) The rational structure QOF∗QOF∗QO_(F)^(**)\mathbf{Q} O_{F}^{*}QOF∗ on the left-hand side corresponds to the structure ζF(r)(0)QXS∞ζF(r)(0)QXS∞zeta_(F)^((r))(0)QX_(S_(oo))\zeta_{F}^{(r)}(0) \mathbf{Q} X_{S_{\infty}}ζF(r)(0)QXS∞ on the right-hand side.
(ii) The integral structure OF∗/μFOF∗/μFO_(F)^(**)//mu_(F)O_{F}^{*} / \mu_{F}OF∗/μF on the left-hand side corresponds to the structure ζF(r)(0)XS∞ζF(r)(0)XS∞zeta_(F)^((r))(0)X_(S_(oo))\zeta_{F}^{(r)}(0) X_{S_{\infty}}ζF(r)(0)XS∞ on the right-hand side.
Motivated by this reformulation, we present Stark's conjecture and its integral refinement due to Rubin. For details, see [41]. Let FFFFF be a number field of degree nnnnn and let H/FH/FH//FH / FH/F be a finite Galois extension with G=Gal(H/F)G=Galâ¡(H/F)G=Gal(H//F)G=\operatorname{Gal}(H / F)G=Galâ¡(H/F) abelian. Let S,TS,TS,TS, TS,T be two finite disjoint sets of places of FFFFF satisfying the following conditions:
(1) SSSSS contains the sets S∞S∞S_(oo)S_{\infty}S∞ of archimedean places and Sram Sram S_("ram ")S_{\text {ram }}Sram of places ramified in HHHHH.
(2) TTTTT contains at least two primes of different residue characteristic or at least one prime of residue characteristic larger than n+1n+1n+1n+1n+1, where n=[F:Q]n=[F:Q]n=[F:Q]n=[F: \mathbf{Q}]n=[F:Q].
For any character χ∈G^=Hom(G,C∗)χ∈G^=Homâ¡G,C∗chi in hat(G)=Hom(G,C^(**))\chi \in \hat{G}=\operatorname{Hom}\left(G, \mathbf{C}^{*}\right)χ∈G^=Homâ¡(G,C∗), define the SSSSS-depleted, TTTTT-smoothed LLLLL-function
The function LS,T(χ,s)LS,T(χ,s)L_(S,T)(chi,s)L_{S, T}(\chi, s)LS,T(χ,s) extends by analytic continuation to a holomorphic function on CCC\mathbf{C}C. The Stickelberger element associated to this data is the unique group-ring element ΘS,T(H/F,s)∈C[G]ΘS,T(H/F,s)∈C[G]Theta_(S,T)(H//F,s)inC[G]\Theta_{S, T}(H / F, s) \in \mathbf{C}[G]ΘS,T(H/F,s)∈C[G] satisfying
χ(ΘS,T(H/F,s))=LS,T(χ−1,s) for all χ∈G^χΘS,T(H/F,s)=LS,Tχ−1,s for all χ∈G^chi(Theta_(S,T)(H//F,s))=L_(S,T)(chi^(-1),s)quad" for all "chi in hat(G)\chi\left(\Theta_{S, T}(H / F, s)\right)=L_{S, T}\left(\chi^{-1}, s\right) \quad \text { for all } \chi \in \hat{G}χ(ΘS,T(H/F,s))=LS,T(χ−1,s) for all χ∈G^
Let SHSHS_(H)S_{H}SH denote the set of places of HHHHH above those in SSSSS, and similarly for THTHT_(H)T_{H}TH. Define
US,T={x∈H∗:ordw(x)≥0 for all w∉SH and x≡1(modTH)}US,T=x∈H∗:ordwâ¡(x)≥0 for all w∉SH and x≡1modTHU_(S,T)={x inH^(**):ord_(w)(x) >= 0" for all "w!inS_(H)" and "x-=1(modT_(H))}U_{S, T}=\left\{x \in H^{*}: \operatorname{ord}_{w}(x) \geq 0 \text { for all } w \notin S_{H} \text { and } x \equiv 1\left(\bmod T_{H}\right)\right\}US,T={x∈H∗:ordwâ¡(x)≥0 for all w∉SH and x≡1(modTH)}
The condition on TTTTT ensures that US,TUS,TU_(S,T)U_{S, T}US,T does not have any torsion. The Galois equivariant version of Dirichlet's unit theorem gives an R[G]R[G]R[G]\mathbf{R}[G]R[G]-module isomorphism
Suppose that exactly rrrrr places v1,…,vr∈Sv1,…,vr∈Sv_(1),dots,v_(r)in Sv_{1}, \ldots, v_{r} \in Sv1,…,vr∈S split completely in HHHHH and #S≥r+1#S≥r+1#S >= r+1\# S \geq r+1#S≥r+1. The order of vanishing of LS,T(χ,s)LS,T(χ,s)L_(S,T)(chi,s)L_{S, T}(\chi, s)LS,T(χ,s) at s=0s=0s=0s=0s=0 is given by
r(χ)=dimC(CUS,T)(χ)={#{v∈S:χ(v)=1} if χ≠1#S−1 if χ=1r(χ)=dimCâ¡CUS,T(χ)=#{v∈S:χ(v)=1} if χ≠1#S−1 if χ=1r(chi)=dim_(C)(CU_(S,T))^((chi))={[#{v in S:chi(v)=1}," if "chi!=1],[#S-1," if "chi=1]:}r(\chi)=\operatorname{dim}_{\mathbf{C}}\left(\mathbf{C} U_{S, T}\right)^{(\chi)}= \begin{cases}\#\{v \in S: \chi(v)=1\} & \text { if } \chi \neq 1 \\ \# S-1 & \text { if } \chi=1\end{cases}r(χ)=dimCâ¡(CUS,T)(χ)={#{v∈S:χ(v)=1} if χ≠1#S−1 if χ=1
whence r(χ)≥rr(χ)≥rr(chi) >= rr(\chi) \geq rr(χ)≥r for all χ∈G^χ∈G^chi in hat(G)\chi \in \hat{G}χ∈G^. Stark's conjecture predicts that the rrrrr th derivative ΘS,T(r)(H/F,0)ΘS,T(r)(H/F,0)Theta_(S,T)^((r))(H//F,0)\Theta_{S, T}^{(r)}(H / F, 0)ΘS,T(r)(H/F,0) captures the "non-rationality" of the map λλlambda\lambdaλ.
Concretely, this states that for each character χχchi\chiχ of GGGGG with r(χ)=rr(χ)=rr(chi)=rr(\chi)=rr(χ)=r, the nonzero number LS,T(r)(χ−1,s)LS,T(r)χ−1,sL_(S,T)^((r))(chi^(-1),s)L_{S, T}^{(r)}\left(\chi^{-1}, s\right)LS,T(r)(χ−1,s) lies in the one-dimensional QQQ\mathbf{Q}Q-vector space spanned by λ(⋀r(US,T(χ)))λ⋀r US,T(χ)lambda(^^^r(U_(S,T)^((chi))))\lambda\left(\bigwedge^{r}\left(U_{S, T}^{(\chi)}\right)\right)λ(â‹€r(US,T(χ))).
Let us reformulate Conjecture 2.1 in terms of the existence of special elements. Write XSH∗=Hom(XSH,Z[G])XSH∗=Homâ¡XSH,Z[G]X_(S_(H))^(**)=Hom(X_(S_(H)),Z[G])X_{S_{H}}^{*}=\operatorname{Hom}\left(X_{S_{H}}, \mathbb{Z}[G]\right)XSH∗=Homâ¡(XSH,Z[G]). For φ∈∧rXSH∗φ∈∧rXSH∗varphi in^^^(r)X_(S_(H))^(**)\varphi \in \wedge^{r} X_{S_{H}}^{*}φ∈∧rXSH∗, there is a determinant map
We extend the determinant map to RRR\mathbf{R}R-linearizations. We fix a place wiwiw_(i)w_{i}wi of HHHHH above each viviv_(i)v_{i}vi. Let wi∗∈XSH∗wi∗∈XSH∗w_(i)^(**)inX_(S_(H))^(**)w_{i}^{*} \in X_{S_{H}}^{*}wi∗∈XSH∗ be induced by
wi∗(w)=∑γ∈G:γwi=wγwi∗(w)=∑γ∈G:γwi=w γw_(i)^(**)(w)=sum_(gamma in G:gammaw_(i)=w)gammaw_{i}^{*}(w)=\sum_{\gamma \in G: \gamma w_{i}=w} \gammawi∗(w)=∑γ∈G:γwi=wγ
Conjecture 2.2 (Stark). Put φ=w1∗∧⋯∧wr∗φ=w1∗∧⋯∧wr∗varphi=w_(1)^(**)^^cdots^^w_(r)^(**)\varphi=w_{1}^{*} \wedge \cdots \wedge w_{r}^{*}φ=w1∗∧⋯∧wr∗. There exists u∈Q∧rUS,Tu∈Q∧rUS,Tu inQ^^^(r)U_(S,T)u \in \mathbf{Q} \wedge^{r} U_{S, T}u∈Q∧rUS,T such that
The equivalence of Conjectures 2.1 and 2.2 is proven in [41, PROPOSITION 2.4].
We are now ready to state the integral version of Stark's conjecture. In the rank r=1r=1r=1r=1r=1 case, Stark proposed the statement that uuuuu in Conjecture 2.2 lies in US,TUS,TU_(S,T)U_{S, T}US,T. This is the famous "rank 1 abelian Stark conjecture." In the higher rank case, the obvious generalization u∈⋀rUS,Tu∈⋀r US,Tu in^^^rU_(S,T)u \in \bigwedge^{r} U_{S, T}u∈⋀rUS,T is not true, as was experimentally observed by Rubin [41]. Rubin defined a lattice, nowadays called "Rubin's lattice" and conjectured that it contains the element uuuuu.
Put US,T∗=HomZ[G](US,T,Z[G])US,T∗=HomZ[G]â¡US,T,Z[G]U_(S,T)^(**)=Hom_(Z[G])(U_(S,T),Z[G])U_{S, T}^{*}=\operatorname{Hom}_{\mathbf{Z}[G]}\left(U_{S, T}, \mathbf{Z}[G]\right)US,T∗=HomZ[G]â¡(US,T,Z[G]).
The rrrrr th exterior bidual of US,TUS,TU_(S,T)U_{S, T}US,T (see [7] for a more general study and the initiation of this terminology) is defined by
⋂rUS,T=(⋀rUS,T∗)∗≅{x∈⋀rQUS,T:φ(x)∈Z[G] for all φ∈⋀rUS,T∗}â‹‚r US,T=â‹€r US,T∗∗≅x∈⋀r QUS,T:φ(x)∈Z[G] for all φ∈⋀r US,T∗nnnrU_(S,T)=(^^^rU_(S,T)^(**))^(**)~={x in^^^rQU_(S,T):varphi(x)inZ[G]" for all "varphi in^^^rU_(S,T)^(**)}\bigcap^{r} U_{S, T}=\left(\bigwedge^{r} U_{S, T}^{*}\right)^{*} \cong\left\{x \in \bigwedge^{r} \mathbf{Q} U_{S, T}: \varphi(x) \in \mathbf{Z}[G] \text { for all } \varphi \in \bigwedge^{r} U_{S, T}^{*}\right\}â‹‚rUS,T=(â‹€rUS,T∗)∗≅{x∈⋀rQUS,T:φ(x)∈Z[G] for all φ∈⋀rUS,T∗}
We would like to consider only the "rank rrrrr " component of this bidual. To this end, for each character χ∈G^χ∈G^chi in hat(G)\chi \in \hat{G}χ∈G^ consider the associated idempotent
We now assume that the totally split places v1,…,vrv1,…,vrv_(1),dots,v_(r)v_{1}, \ldots, v_{r}v1,…,vr from the previous section are all finite. This happens only when FFFFF is a totally real field and HHHHH is totally complex. In fact, the fixed fields of characters with nonvanishing LLLLL-functions at 0 are CMCMCM\mathrm{CM}CM fields, so we restrict to the setting where FFFFF is totally real and HHHHH is CMCMCM\mathrm{CM}CM for the remainder of the article. We also
enact a slight notational change and write the set denoted SSSSS in the previous sections as S′S′S^(')S^{\prime}S′, and let S=S′∖{v1,…,vr}S=S′∖v1,…,vrS=S^(')\\{v_(1),dots,v_(r)}S=S^{\prime} \backslash\left\{v_{1}, \ldots, v_{r}\right\}S=S′∖{v1,…,vr}. The reason for this is that we now still have S⊃S∞∪Sram S⊃S∞∪Sram S supS_(oo)uuS_("ram ")S \supset S_{\infty} \cup S_{\text {ram }}S⊃S∞∪Sram .
As we explain, in this setting Conjecture 2.2 for S′S′S^(')S^{\prime}S′ follows from a classical rationality result of Klingen-Siegel, though the integral refinement in Conjecture 2.3 remains a nontrivial statement. For a fixed place wwwww of HHHHH, we have
(3.1)log|u|w=−ordw(u)logNw(3.1)logâ¡|u|w=−ordwâ¡(u)logâ¡Nw{:(3.1)log |u|_(w)=-ord_(w)(u)log Nw:}\begin{equation*}
\log |u|_{w}=-\operatorname{ord}_{w}(u) \log \mathrm{N} w \tag{3.1}
\end{equation*}(3.1)logâ¡|u|w=−ordwâ¡(u)logâ¡Nw
Since the Euler factors at the viviv_(i)v_{i}vi are equal to (1−Nvi−s)=(1−Nwi−s)1−Nvi−s=1−Nwi−s(1-Nv_(i)^(-s))=(1-Nw_(i)^(-s))\left(1-\mathrm{N} v_{i}^{-s}\right)=\left(1-\mathrm{N} w_{i}^{-s}\right)(1−Nvi−s)=(1−Nwi−s), we also have
Theorem 3.1 (Klingen-Siegel). We have ΘS,T:=ΘS,T(H/F,0)∈Q[G]ΘS,T:=ΘS,T(H/F,0)∈Q[G]Theta_(S,T):=Theta_(S,T)(H//F,0)inQ[G]\Theta_{S, T}:=\Theta_{S, T}(H / F, 0) \in \mathbf{Q}[G]ΘS,T:=ΘS,T(H/F,0)∈Q[G].
With erere_(r)e_{r}er as in the previous section, we are then led to define a map over QQQ\mathbf{Q}Q
λQ:er(QUS′,T)→er(QXSH′),λQ(u)=∑w∣vi some iordw(u)⋅wλQ:erQUS′,T→erQXSH′,λQ(u)=∑w∣vi some i ordwâ¡(u)â‹…wlambda_(Q):e_(r)(QU_(S^('),T))rarre_(r)(QX_(S_(H)^('))),quadlambda_(Q)(u)=sum_(w∣v_(i)" some "i)ord_(w)(u)*w\lambda_{\mathbf{Q}}: e_{r}\left(\mathbf{Q} U_{S^{\prime}, T}\right) \rightarrow e_{r}\left(\mathbf{Q} X_{S_{H}^{\prime}}\right), \quad \lambda_{\mathbf{Q}}(u)=\sum_{w \mid v_{i} \text { some } i} \operatorname{ord}_{w}(u) \cdot wλQ:er(QUS′,T)→er(QXSH′),λQ(u)=∑w∣vi some iordwâ¡(u)â‹…w
Note that er(QXSH′)erQXSH′e_(r)(QX_(S_(H)^(')))e_{r}\left(\mathbf{Q} X_{S_{H}^{\prime}}\right)er(QXSH′) is the QQQ\mathbf{Q}Q-vector space generated by the places of HHHHH above the viviv_(i)v_{i}vi. The map λQλQlambda_(Q)\lambda_{\mathbf{Q}}λQ is a Q[G]Q[G]Q[G]\mathbf{Q}[G]Q[G]-module isomorphism, and it induces an isomorphism on the free rank one Q[G]Q[G]Q[G]\mathbf{Q}[G]Q[G]-modules obtained by taking rrrrr th wedge powers. In view of (3.1), the map on rrrrr th wedge powers induced by the map λλlambda\lambdaλ of (2.2), when restricted to er(Q∧rUS′,T)erQ∧rUS′,Te_(r)(Q^^^(r)U_(S^('),T))e_{r}\left(\mathbf{Q} \wedge^{r} U_{S^{\prime}, T}\right)er(Q∧rUS′,T), is equal to (∏i=1rlogNwi)⋅λQâˆi=1r logâ¡Nwi⋅λQ(prod_(i=1)^(r)log Nw_(i))*lambda_(Q)\left(\prod_{i=1}^{r} \log \mathrm{N} w_{i}\right) \cdot \lambda_{\mathbf{Q}}(âˆi=1rlogâ¡Nwi)⋅λQ. Conjecture 2.2 follows from this observation together with (3.2), since Theorem 3.1 implies the existence of u∈er(Q⋀rUS′,T)u∈erQâ‹€r US′,Tu ine_(r)(Q^^^rU_(S^('),T))u \in e_{r}\left(\mathbf{Q} \bigwedge^{r} U_{S^{\prime}, T}\right)u∈er(Qâ‹€rUS′,T) such that
Here φ=w1∗∧⋯∧wr∗φ=w1∗∧⋯∧wr∗varphi=w_(1)^(**)^^cdots^^w_(r)^(**)\varphi=w_{1}^{*} \wedge \cdots \wedge w_{r}^{*}φ=w1∗∧⋯∧wr∗ as in the statement of the conjecture.
On the other hand, the integral statement in Conjecture 2.3 lies deeper. We first note the following celebrated theorem of Deligne-Ribet [21] and Cassou-Noguès [8] refining the Klingen-Siegel theorem. The condition on the set TTTTT stated in Section 2 is crucial in this result. We remark that Deligne-RIbet prove their result using Hilbert modular forms, as an integral refinement of the strategy of the strategy established earlier by Siegel. This theme reappears in our own work described in §6§6§6\S 6§§6.
Theorem 3.2. We have ΘS,T∈Z[G]ΘS,T∈Z[G]Theta_(S,T)inZ[G]\Theta_{S, T} \in \mathbf{Z}[G]ΘS,T∈Z[G].
Conjecture 2.3 in this setting is known as the Rubin-Brumer-Stark conjecture:
Conjecture 3.3 (Rubin-Brumer-Stark). There exists u∈L(r)US′,Tu∈L(r)US′,Tu inL^((r))U_(S^('),T)u \in \mathscr{L}^{(r)} U_{S^{\prime}, T}u∈L(r)US′,T such that
We describe in Theorem 4.3 below a strong partial result toward this conjecture.
4. THE BRUMER-STARK CONJECTURE
Having stated the higher rank Rubin-Brumer-Stark conjecture, we now wind back the clock and focus on the case r=1r=1r=1r=1r=1. This case had been studied independently by Brumer and Stark and served as a motivation for Rubin's work. Writing the splitting prime v1v1v_(1)v_{1}v1 as ppp\mathfrak{p}p, the conjecture may be stated as follows.
Conjecture 4.1 (Brumer-Stark). Fix a prime ideal p⊂OF,p∉S∪Tp⊂OF,p∉S∪TpsubO_(F),p!in S uu T\mathfrak{p} \subset O_{F}, \mathfrak{p} \notin S \cup Tp⊂OF,p∉S∪T, such that ppp\mathfrak{p}p splits completely in HHHHH. Fix a prime P⊂OHP⊂OHPsubO_(H)\mathfrak{P} \subset O_{H}P⊂OH above ppp\mathfrak{p}p. There exists a unique element up∈H∗up∈H∗u_(p)inH^(**)u_{\mathfrak{p}} \in H^{*}up∈H∗ such that |up|w=1upw=1|u_(p)|_(w)=1\left|u_{\mathfrak{p}}\right|_{w}=1|up|w=1 for every place wwwww of HHHHH not lying above ppp\mathfrak{p}p,
ordG(up):=∑σ∈Gordβ(σ(up))σ−1=ΘS,TordGâ¡up:=∑σ∈G ordβâ¡Ïƒupσ−1=ΘS,Tord_(G)(u_(p)):=sum_(sigma in G)ord_(beta)(sigma(u_(p)))sigma^(-1)=Theta_(S,T)\operatorname{ord}_{G}\left(u_{\mathfrak{p}}\right):=\sum_{\sigma \in G} \operatorname{ord}_{\mathfrak{\beta}}\left(\sigma\left(u_{\mathfrak{p}}\right)\right) \sigma^{-1}=\Theta_{S, T}ordGâ¡(up):=∑σ∈Gordβâ¡(σ(up))σ−1=ΘS,T
and u≡1(modq)u≡1(modq)u-=1(modq)u \equiv 1(\bmod \mathfrak{q})u≡1(modq) for all q∈THq∈THqinT_(H)\mathfrak{q} \in T_{H}q∈TH.
Note that the condition |u|w=1|u|w=1|u|_(w)=1|u|_{w}=1|u|w=1 includes all complex places wwwww, so c(up)=up−1cup=up−1c(u_(p))=u_(p)^(-1)c\left(u_{\mathfrak{p}}\right)=u_{\mathfrak{p}}^{-1}c(up)=up−1 for the unique complex conjugation c∈Gc∈Gc in Gc \in Gc∈G.
As we have alread noted, Stark arrived upon this statement in the 1970s through his attempts to generalize and factorize the classical Dirichlet class number formula (though in a slightly different formulation; the statement above is due to Tate [46]). Prior to this, in the 1960s, Brumer was interested in generalizing Stickelberger's classical factorization formula for Gauss sums in cyclotomic fields. Stickelberger's result can be formulated as stating that when H=Q(μN)H=QμNH=Q(mu_(N))H=\mathbf{Q}\left(\mu_{N}\right)H=Q(μN) is a cyclotomic field, the Stickelberger element annihilates the class group of HHHHH. Let us consider Brumer's perspective of annihilation of class groups in the case of general H/FH/FH//FH / FH/F.
4.1. Annihilation of class groups
Let ClT(H)ClT(H)Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H) denote the ray class group of HHHHH with conductor equal to the product of primes in THTHT_(H)T_{H}TH. This is defined as follows. Let IT(H)IT(H)I_(T)(H)I_{T}(H)IT(H) denote the group of fractional ideals of HHHHH relatively prime to the primes in THTHT_(H)T_{H}TH. Let PT(H)PT(H)P_(T)(H)P_{T}(H)PT(H) denote the subgroup of IT(H)IT(H)I_(T)(H)I_{T}(H)IT(H) generated by principal ideals (α)(α)(alpha)(\alpha)(α) where α∈OHα∈OHalpha inO_(H)\alpha \in O_{H}α∈OH satisfies α≡1(modq)α≡1(modq)alpha-=1(mod q)\alpha \equiv 1(\bmod q)α≡1(modq) for all q∈THq∈THq inT_(H)q \in T_{H}q∈TH. Then
Such an equation holds for all p∉S∪Tp∉S∪Tp!in S uu Tp \notin S \cup Tp∉S∪T that split completely in HHHHH. The set of primes of HHHHH above all such ppp\mathfrak{p}p generate ClT(H)ClT(H)Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H). Hence we deduce
In fact, (4.3) is almost equivalent to Conjecture 4.1; given (4.2), the element upupu_(p)u_{\mathfrak{p}}up satisfies all of the conditions necessary for Conjecture 4.1 except possibly c(up)=up−1cup=up−1c(u_(p))=u_(p)^(-1)c\left(u_{\mathfrak{p}}\right)=u_{\mathfrak{p}}^{-1}c(up)=up−1. But of course vp=up/c(up)vp=up/cupv_(p)=u_(p)//c(u_(p))v_{\mathfrak{p}}=u_{\mathfrak{p}} / c\left(u_{\mathfrak{p}}\right)vp=up/c(up) satisfies this condition and moreover satisfies P2ΘS,T=(vp)P2ΘS,T=vpP^(2Theta_(S,T))=(v_(p))\mathfrak{P}^{2 \Theta_{S, T}}=\left(v_{\mathfrak{p}}\right)P2ΘS,T=(vp). Therefore the
only possible discrepancy between the statements is a factor of 2 , which disappears when we localize away from 2 as in the rest of this paper. Let us therefore define
and for any Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G]-module MMMMM we write M−=M⊗Z[G]RM−=M⊗Z[G]RM^(-)=Mox_(Z[G])RM^{-}=M \otimes_{\mathbf{Z}[G]} RM−=M⊗Z[G]R. There exists an element up∈OH[1/p]∗⊗Z[1/2]up∈OH[1/p]∗⊗Z[1/2]u_(p)inO_(H)[1//p]^(**)oxZ[1//2]u_{\mathfrak{p}} \in O_{H}[1 / \mathfrak{p}]^{*} \otimes \mathbf{Z}[1 / 2]up∈OH[1/p]∗⊗Z[1/2] satisfying Conjecture 4.1 if and only if
This is the Brumer-Stark conjecture "away from 2".
Many authors have studied (4.4) as well as refinements. The works of Burns, Greither, Kurihara, Popescu, and Sano are particularly noteworthy [4-7, 25-27,35]. Many of these refinements involve Fitting ideals, whose definition we now recall.
Let RRRRR be a commutative ring and MMMMM an RRRRR-module with finite presentation:
Rm→ARn→X→0Rm→ARn→X→0R^(m)rarr"A"R^(n)rarr X rarr0R^{m} \xrightarrow{A} R^{n} \rightarrow X \rightarrow 0Rm→ARn→X→0
Here AAAAA is an n×mn×mn xx mn \times mn×m matrix over RRRRR. The iiiii th Fitting ideal Fitt i,R(M)i,R(M)i,R(M)i, R(M)i,R(M) is the ideal generated by the n−i×n−in−i×n−in-i xx n-in-i \times n-in−i×n−i minors of AAAAA. It is a standard fact [37, CHAPTER 3, THEOREM 1] that Fitt i,R(M)i,R(M)_(i,R)(M){ }_{i, R}(M)i,R(M) does not depend on the chosen presentation of MMMMM. We write FittR(M)FittRâ¡(M)Fitt_(R)(M)\operatorname{Fitt}_{R}(M)FittRâ¡(M) for Fitt0,R(M)Fitt0,Râ¡(M)Fitt_(0,R)(M)\operatorname{Fitt}_{0, R}(M)Fitt0,Râ¡(M), and when these is no ambiguity about the choice of i=0i=0i=0i=0i=0, we call this the Fitting ideal of MMMMM.
The Fitting ideal of a finitely presented module is contained in its annihilator:
In view of (4.4) and (4.5), it is therefore natural to ask whether ΘS,TΘS,TTheta_(S,T)\Theta_{S, T}ΘS,T lies in the Fitting ideal of ClT(H)−ClT(H)−Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−over RRRRR. It was noticed by Popescu in the function field case [38] and by Kurihara in the number field case that while this holds sometimes, it does not always hold. Greither and Kurihara [25,26][25,26][25,26][25,26][25,26] observed that the statement may be corrected by replacing ClT(H)−ClT(H)−Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−by its Pontryagin dual
We endow ClT(H)−,∨ClT(H)−,∨Cl^(T)(H)^(-,vv)\mathrm{Cl}^{T}(H)^{-, \vee}ClT(H)−,∨ with the contragradient GGGGG-action g⋅φ(x)=φ(g−1x)g⋅φ(x)=φg−1xg*varphi(x)=varphi(g^(-1)x)g \cdot \varphi(x)=\varphi\left(g^{-1} x\right)g⋅φ(x)=φ(g−1x). Denote by # the involution on Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G] induced by g↦g−1g↦g−1g|->g^(-1)g \mapsto g^{-1}g↦g−1 for g∈Gg∈Gg in Gg \in Gg∈G.
Conjecture 4.2 (Kurihara, "Strong Brumer-Stark"). We have
Conjecture 4.2 leads to the following natural questions:
(1) What is the Fitting ideal of ClT(H)−,vClT(H)−,vCl^(T)(H)^(-,v)\mathrm{Cl}^{T}(H)^{-, v}ClT(H)−,v ?
(2) What is the Fitting ideal of ClT(H)−ClT(H)−Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−?
(3) Is there a natural arithmetically defined RRRRR-module whose Fitting ideal is generated by ΘS,TΘS,TTheta_(S,T)\Theta_{S, T}ΘS,T or ΘS,T#ΘS,T#Theta_(S,T)^(#)\Theta_{S, T}^{\#}ΘS,T# ?
The precise conjectural description of the Fitting ideal of ClT(H/F)−,∨ClT(H/F)−,∨Cl^(T)(H//F)^(-,vv)\mathrm{Cl}^{T}(H / F)^{-, \vee}ClT(H/F)−,∨ was given by Kurihara [35]; we state this in Section 5.1 below. An important fact about this statement is that when Sram Sram S_("ram ")S_{\text {ram }}Sram is nonempty, the Fitting ideal of ClT(H/F)−,∨ClT(H/F)−,∨Cl^(T)(H//F)^(-,vv)\mathrm{Cl}^{T}(H / F)^{-, \vee}ClT(H/F)−,∨ is in general not principal (and in particular is not generated by ΘS,T#ΘS,T#Theta_(S,T)^(#)\Theta_{S, T}^{\#}ΘS,T# ).
A conjectural answer to the second question above has recently been provided in a striking paper by Atsuta and Kataoka [1]. They show that their conjecture is implied by the Equivariant Tamagawa Number Conjecture.
The third question is answered by a conjecture of Burns, Kurihara, and Sano, and is the topic of Section §5.3§5.3§5.3\S 5.3§§5.3. We note that Fitting ideals of finitely presented RRRRR-modules are rarely principal. It is therefore remarkable that Burns-Kurihara-Sano defined a natural arithmetic object whose Fitting ideal is principal.
4.2. Our results
We now describe some of our results toward these conjectures [17, THEOREM 1.4].
Theorem 4.3. Kurihara's exact formula for FittR(ClT(H)−,∨)FittRClT(H)−,∨Fitt_(R)(Cl^(T)(H)^(-,vv))\mathrm{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-, \vee}\right)FittR(ClT(H)−,∨) holds (see Theorem 5.1). In particular, we have the Brumer-Stark and Strong Brumer-Stark conjectures away from 2:
Finally, Rubin's higher rank Brumer-Stark conjecture holds away from 2: with notation as in Conjecture 3.3, there exists u∈L(r)US′,T⊗Z[1/2]u∈L(r)US′,T⊗Z[1/2]u inL^((r))U_(S^('),T)oxZ[1//2]u \in \mathscr{L}^{(r)} U_{S^{\prime}, T} \otimes \mathbf{Z}[1 / 2]u∈L(r)US′,T⊗Z[1/2] such that φ(λQ(u))=ΘS,TφλQ(u)=ΘS,Tvarphi(lambda_(Q)(u))=Theta_(S,T)\varphi\left(\lambda_{\mathbf{Q}}(u)\right)=\Theta_{S, T}φ(λQ(u))=ΘS,T.
Partial results in this direction had been obtained earlier by Burns [5] (including a μ=0μ=0mu=0\mu=0μ=0 hypothesis and the assumption of the Gross-Kuz'min conjecture) and by Greither and Popescu [27] (including a μ=0μ=0mu=0\mu=0μ=0 hypothesis and imprimitivity conditions on SSSSS ).
Our results in [17] do not seem to directly imply the conjecture of Atsuta and Kataoka on Fitt R(ClT(H)−)RClT(H)−_(R)(Cl^(T)(H)^(-)){ }_{R}\left(\mathrm{Cl}^{T}(H)^{-}\right)R(ClT(H)−)or the conjecture of Burns. However, we prove an analogous result toward the latter, with (S,T)(S,T)(S,T)(S, T)(S,T) replaced by an alternate pair (Σ,Σ′)Σ,Σ′(Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′), in Theorem 5.6. This result turns out to be strong enough to deduce Theorem 4.3.
In §6§6§6\S 6§§6 we give a detailed summary of the proof of Theorem 5.6. Key ingredients are the Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G]-modules ∇ΣΣ′(H)∇ΣΣ′(H)grad_(Sigma)^(Sigma^('))(H)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)∇ΣΣ′(H) defined by Ritter and Weiss, and Ribet's method of using modular forms to construct Galois cohomology classes associated to LLLLL-functions.
4.3. Explicit formula for Brumer-Stark units
We conclude this section by describing a further direction in the study of BrumerStark units, that of explicit formulae and applications to explicit class field theory. This theme was initiated by Gross in [28] and [29] and developed by the first author and collaborators over a series of papers [9,10,13,15,20][9,10,13,15,20][9,10,13,15,20][9,10,13,15,20][9,10,13,15,20].
Let ppp\mathfrak{p}p be as above and write S′=S∪{p}S′=S∪{p}S^(')=S uu{p}S^{\prime}=S \cup\{\mathfrak{p}\}S′=S∪{p}. Let LLLLL denote a finite abelian CM extension of FFFFF containing HHHHH that is ramified over FFFFF only at the places in S′S′S^(')S^{\prime}S′. Write g=Gal(L/F)g=Galâ¡(L/F)g=Gal(L//F)\mathrm{g}=\operatorname{Gal}(L / F)g=Galâ¡(L/F) and Γ=Gal(L/H)Γ=Galâ¡(L/H)Gamma=Gal(L//H)\Gamma=\operatorname{Gal}(L / H)Γ=Galâ¡(L/H), so g/Γ≅Gg/Γ≅Gg//Gamma~=G\mathrm{g} / \Gamma \cong Gg/Γ≅G. Let IIIII denote the relative augmentation ideal associated to ggggg and GGGGG, i.e., the kernel of the canonical projection Aug: Z[g]→Z[G]Z[g]→Z[G]Z[g]rarrZ[G]\mathbf{Z}[\mathfrak{g}] \rightarrow \mathbf{Z}[G]Z[g]→Z[G]. Then ΘS′,T(L/F)ΘS′,T(L/F)Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F)
lies in IIIII, since its image under Aug is
Here σpσpsigma_(p)\sigma_{\mathfrak{p}}σp denotes the Frobenius associated to ppp\mathfrak{p}p in GGGGG, and this is trivial since ppp\mathfrak{p}p splits completely in HHHHH. Intuitively, if we view ΘS′,T(L/F)ΘS′,T(L/F)Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F) as a function on the ideals of Z[g]Z[g]Z[g]\mathbf{Z}[\mathrm{g}]Z[g], equation (4.6) states that this function "has a zero" at the ideal IIIII; the value of the "derivative" of this function at IIIII is simply the image of ΘS′,T(L/F)ΘS′,T(L/F)Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F) in I/I2I/I2I//I^(2)I / I^{2}I/I2. Gross provided a conjectural algebraic interpretation of this derivative as follows. Denote by
the composition of the inclusion Hß∗↪AH∗Hß∗↪AH∗H_(ß)^(**)↪A_(H)^(**)H_{\mathfrak{ß}}^{*} \hookrightarrow \mathbf{A}_{H}^{*}ßHß∗↪AH∗ with the global Artin reciprocity map
Throughout this article we adopt Serre's convention [42] for the reciprocity map. Therefore rec(ϖ−1)recâ¡Ï–−1rec(Ï–^(-1))\operatorname{rec}\left(\varpi^{-1}\right)recâ¡(ϖ−1) is a lifting to GpabGpabG_(p)^(ab)G_{\mathfrak{p}}^{\mathrm{ab}}Gpab of the Frobenius element on the maximal unramified extension of FpFpF_(p)F_{\mathfrak{p}}Fp if ϖ∈Fp∗ϖ∈Fp∗ϖinF_(p)^(**)\varpi \in F_{\mathfrak{p}}^{*}ϖ∈Fp∗ is a uniformizer.
(4.7)recG(up)=∑σ∈G(recpσ(up)−1)σ~−1∈I/I2(4.7)recGâ¡up=∑σ∈G recpâ¡Ïƒup−1σ~−1∈I/I2{:(4.7)rec_(G)(u_(p))=sum_(sigma in G)(rec_(p)sigma(u_(p))-1) tilde(sigma)^(-1)in I//I^(2):}\begin{equation*}
\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right)=\sum_{\sigma \in G}\left(\operatorname{rec}_{\mathfrak{p}} \sigma\left(u_{\mathfrak{p}}\right)-1\right) \tilde{\sigma}^{-1} \in I / I^{2} \tag{4.7}
\end{equation*}(4.7)recGâ¡(up)=∑σ∈G(recpâ¡Ïƒ(up)−1)σ~−1∈I/I2
where σ~∈gσ~∈gtilde(sigma)ing\tilde{\sigma} \in \mathrm{g}σ~∈g is any lift of σ∈Gσ∈Gsigma in G\sigma \in Gσ∈G. Then
recG(up)≡ΘS′,TL/F in I/I2recGâ¡up≡ΘS′,TL/F in I/I2rec_(G)(u_(p))-=Theta_(S^('),T)^(L//F)quad" in "I//I^(2)\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right) \equiv \Theta_{S^{\prime}, T}^{L / F} \quad \text { in } I / I^{2}recGâ¡(up)≡ΘS′,TL/F in I/I2
The main result of [18] is the following.
Theorem 4.5. Let ppppp be an odd prime and suppose that ppp\mathrm{p}p lies above p. Gross's Conjecture 4.4 holds in (I/I2)⊗ZpI/I2⊗Zp(I//I^(2))oxZ_(p)\left(I / I^{2}\right) \otimes \mathbf{Z}_{p}(I/I2)⊗Zp.
Our interest in this result is that by enlarging SSSSS and taking larger and larger field extensions L/FL/FL//FL / FL/F, one can use (4.7) to specify all of the ppp\mathfrak{p}p-adic digits of upupu_(p)u_{\mathfrak{p}}up. One therefore obtains an exact ppp\mathfrak{p}p-adic analytic formula for upupu_(p)u_{\mathfrak{p}}up. This formula can be described either using the Eisenstein cocycle or more explicitly via Shintani's method; the latter approach is followed in Section 7.2. In Section 7.3, we describe the argument using "horizontal Iwasawa theory" to show that Theorem 4.5 implies the conjectural exact formula. In Section 7.4 we summarize the key ingredients involved in the proof of Theorem 4.5, including an integral version of the Greenberg-Stevens LLL\mathscr{L}L-invariant and an associated modified Ritter-Weiss module ∇L∇Lgrad_(L)\nabla_{\mathscr{L}}∇L. In the setting of FFFFF real quadratic, Darmon, Pozzi, and Vonk have given an alternate, elegant proof of the explicit formula for the units upupu_(p)u_{\mathfrak{p}}up (Section 7.5). Their approach involves ppppp-adic deformations of modular forms, rather than the tame deformations that we consider.
One significance of the exact formula is that we show that the collection of BrumerStark units, together with some easily described elements, generate the maximal abelian extension of the totally real field FFFFF.
Theorem 4.6. Let BS denote the set of Brumer-Stark units upupu_(p)u_{\mathfrak{p}}up as we range over all possible CMabelian extensions H/FH/FH//FH / FH/F and for each extension a choice of prime ppppp that splits completely in HHHHH. Let {α1,…,αn−1}α1,…,αn−1{alpha_(1),dots,alpha_(n-1)}\left\{\alpha_{1}, \ldots, \alpha_{n-1}\right\}{α1,…,αn−1} denote any elements of F∗F∗F^(**)F^{*}F∗ whose signs in {±1}n/(−1,…,−1){±1}n/(−1,…,−1){+-1}^(n)//(-1,dots,-1)\{ \pm 1\}^{n} /(-1, \ldots,-1){±1}n/(−1,…,−1) under the real embeddings of FFFFF form a basis for this Z/2ZZ/2ZZ//2Z\mathbf{Z} / 2 \mathbf{Z}Z/2Z-vector space. The maximal abelian extension of FFFFF is generated by BSBSBS\mathrm{BS}BS together with α1,…,αn−1α1,…,αn−1sqrt(alpha_(1)),dots,sqrt(alpha_(n-1))\sqrt{\alpha_{1}}, \ldots, \sqrt{\alpha_{n-1}}α1,…,αn−1 :
It is important to stress that the exact formula for upupu_(p)u_{\mathfrak{p}}up described in Sectin 7.2 can be computed without knowledge of the field HHHHH, using only the data of FFFFF, p, and the conductor of H/FH/FH//FH / FH/F. Furthermore, we can leave out any finite set of primes ppp\mathfrak{p}p without altering the conclusion of the theorem. In this way we obtain an explicit class field theory for FFFFF, i.e., an analytic construction of its maximal abelian extension Fab Fab F^("ab ")F^{\text {ab }}Fab using data intrinsic only to FFFFF itself. Explicit computations of class fields of real quadratic fields generated using our formula are provided in [17, §2.3] and [23].
5. REFINEMENTS OF STARK'S CONJECTURE
In this section we recall various refinements of the strong Brumer-Stark conjecture.
5.1. The conjecture of Kurihara
In this section we describe the Fitting ideal of the minus part of the dual class group. For each vvvvv in Sram Sram S_("ram ")S_{\text {ram }}Sram , let Iv⊂Gv⊂GIv⊂Gv⊂GI_(v)subG_(v)sub GI_{v} \subset G_{v} \subset GIv⊂Gv⊂G denote the inertia and decomposition groups, respectively, associated to vvvvv. Write
for the idempotent that represents projection onto the characters of GGGGG unramified at vvvvv. Let σv∈Gvσv∈Gvsigma_(v)inG_(v)\sigma_{v} \in G_{v}σv∈Gv denote any representative of the Frobenius coset of vvvvv. The element 1−σvev∈Q[G]1−σvev∈Q[G]1-sigma_(v)e_(v)inQ[G]1-\sigma_{v} e_{v} \in \mathbf{Q}[G]1−σvev∈Q[G] is independent of choice of representative. Following [25], we define the Sinnott-Kurihara ideal, a priori a fractional ideal of Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G], by
Kurihara proved using the theorem of Deligne-Ribet and Cassou-Noguès that SKuT(H/F)SKuT(H/F)SKu^(T)(H//F)\mathrm{SKu}^{T}(H / F)SKuT(H/F) is a subset of Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G] (see [17, LEMMA 3.4]). The following conjecture of Kurihara is proven in [17[17[17[17[17, THEOREM 1.4].
The definition of the Sinnott-Kurihara ideal is inspired by Sinnott's definition of generalized Stickelberger elements for abelian extensions of QQQ\mathbf{Q}Q [44]. For a generalization of Sinnott's ideal to arbitrary totally real fields see [25, $2]. In general, Sinnott's ideal contains the Sinnott-Kurihara ideal but it may be strictly larger.
The plus part of the Sinnott-Kurihara ideal is not very interesting as the plus part of ΘS∞,T#ΘS∞,T#Theta_(S_(oo),T)^(#)\Theta_{S_{\infty}, T}^{\#}ΘS∞,T# is 0 . The plus part of the class group is much smaller than the minus part and seems harder to describe; for example, Greenberg's conjecture on the vanishing of lambda invariants implies that the order of the plus part is bounded up the cyclotomic tower. For abelian extensions of QQQ\mathbf{Q}Q, the plus part is described by Sinnott using cyclotomic units.
5.2. The conjecture of Atsuta-Kataoka
It is, in fact, more natural to ask about the Fitting ideal of ClT(H)ClT(H)Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H), as opposed to the Pontryagin dual. A conjectural answer to this question has been provided in a recent paper of Atsuta-Kataoka [1] using the theory of shifted Fitting ideals developed by Kataoka [32]. We recall this notion now.
Let MMMMM be an RRRRR-module of finite length. Take a resolution
0→N→P1→⋯→Pd→M→00→N→P1→⋯→Pd→M→00rarr N rarrP_(1)rarr cdots rarrP_(d)rarr M rarr00 \rightarrow N \rightarrow P_{1} \rightarrow \cdots \rightarrow P_{d} \rightarrow M \rightarrow 00→N→P1→⋯→Pd→M→0
with each PiPiP_(i)P_{i}Pi of projective dimension ≤1≤1<= 1\leq 1≤1. Following [32], define the shifted Fitting ideal
In [1], the authors give an explicit description of the ideal hv−FittZ[G]−[1](Av−)hv−FittZ[G]−[1]â¡Av−h_(v)^(-)Fitt_(Z[G]^(-))^([1])(A_(v)^(-))h_{v}^{-} \operatorname{Fitt}_{\mathrm{Z}[G]^{-}}^{[1]}\left(A_{v}^{-}\right)hv−FittZ[G]−[1]â¡(Av−)appearing in Conjecture 5.2. Write Iv=J1×⋯×JsIv=J1×⋯×JsI_(v)=J_(1)xx cdots xxJ_(s)I_{v}=J_{1} \times \cdots \times J_{s}Iv=J1×⋯×Js for cyclic groups Ji,1≤i≤sJi,1≤i≤sJ_(i),1 <= i <= sJ_{i}, 1 \leq i \leq sJi,1≤i≤s. For each iiiii, put
Furthermore, put d=ker(Z[G]→Z[G/Gv])d=kerâ¡Z[G]→ZG/Gvd=ker(Z[G]rarrZ[G//G_(v)])d=\operatorname{ker}\left(\mathbf{Z}[G] \rightarrow \mathbf{Z}\left[G / G_{v}\right]\right)d=kerâ¡(Z[G]→Z[G/Gv]) for the relative augmentation ideal. For each 1≤i≤s1≤i≤s1 <= i <= s1 \leq i \leq s1≤i≤s, put ZiZiZ_(i)Z_{i}Zi for the ideal of Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G] generated by Nj1⋯Njs−iNj1⋯Njs−iN_(j_(1))cdotsN_(j_(s-i))\mathrm{N}_{j_{1}} \cdots \mathrm{N}_{j_{s-i}}Nj1⋯Njs−i, where (j1,…,js−i)j1,…,js−i(j_(1),dots,j_(s-i))\left(j_{1}, \ldots, j_{s-i}\right)(j1,…,js−i) runs through all tuples of integers satisfying 1≤j1≤⋯≤js−i≤s1≤j1≤⋯≤js−i≤s1 <= j_(1) <= cdots <= j_(s-i) <= s1 \leq j_{1} \leq \cdots \leq j_{s-i} \leq s1≤j1≤⋯≤js−i≤s. Define
Although ZiZiZ_(i)Z_{i}Zi depends on the decomposition of IvIvI_(v)I_{v}Iv into cyclic groups, the ideal LLL\mathcal{L}L is independent (see [1, DEFINITION 1.2]).
as fractional ideals of Z[G]−Z[G]−Z[G]^(-)\mathbf{Z}[G]^{-}Z[G]−.
Atsuta-Kataoka prove:
Theorem 5.4. The Equivariant Tamagawa Number Conjecture for H/FH/FH//FH / FH/F implies Conjecture 5.2.
5.3. The conjecture of Burns-Kurihara-Sano
The refinements mentioned above do not involve principal ideals. The method of Ribet, which attempts to show the inclusion of an arithmetically defined ideal into an analytically defined ideal, works well for principal ideals. From this point of view, it is natural to ask if there is an arithmetically defined object whose Fitting ideal is generated by the Stickelberger element ΘS,TΘS,TTheta_(S,T)\Theta_{S, T}ΘS,T. Burns, Kurihara and Sano provided a conjectural answer to this question [6]. A modification of this statement (Theorem 5.6 below) is the main technical result in [17] from which all the results of Theorem 4.3 are deduced.
We now recall the statement of the conjecture of Burns-Kurihara-Sano. Let HT∗HT∗H_(T)^(**)H_{T}^{*}HT∗ be the group of x∈H∗x∈H∗x inH^(**)x \in H^{*}x∈H∗ such that ordw(x−1)>0ordwâ¡(x−1)>0ord_(w)(x-1) > 0\operatorname{ord}_{w}(x-1)>0ordwâ¡(x−1)>0 for each prime w∈THw∈THw inT_(H)w \in T_{H}w∈TH. Define
where the implicit map sends a tuple (xw)xw(x_(w))\left(x_{w}\right)(xw) to the function ∑wxwordw∑w xwordwsum_(w)x_(w)ord_(w)\sum_{w} x_{w} \operatorname{ord}_{w}∑wxwordw. The GGGGG-action on SelST(H)SelSTâ¡(H)Sel_(S)^(T)(H)\operatorname{Sel}_{S}^{T}(H)SelSTâ¡(H) is the contragradient GGGGG-action (gφ)(x)=φ(g−1x)(gφ)(x)=φg−1x(g varphi)(x)=varphi(g^(-1)x)(g \varphi)(x)=\varphi\left(g^{-1} x\right)(gφ)(x)=φ(g−1x)
We have proven a version of this result with altered sets SSSSS and TTTTT. Fix an odd prime ppppp and put Rp=Zp[G]−Rp=Zp[G]−R_(p)=Z_(p)[G]^(-)R_{p}=\mathbf{Z}_{p}[G]^{-}Rp=Zp[G]−. Define
Σ=S∖{v∈S:v∤p}Σ=S∖{v∈S:v∤p}Sigma=S\\{v in S:v∤p}\Sigma=S \backslash\{v \in S: v \nmid p\}Σ=S∖{v∈S:v∤p}
and
Σ′=T∪{v∈S:v∤p}Σ′=T∪{v∈S:v∤p}Sigma^(')=T uu{v in S:v∤p}\Sigma^{\prime}=T \cup\{v \in S: v \nmid p\}Σ′=T∪{v∈S:v∤p}
Theorem 5.6 ([17, THEOREM 3.3]). Let SelΣΣ′(H)p−=SelΣΣ′(H)⊗Z[G]RpSelΣΣ′â¡(H)p−=SelΣΣ′â¡(H)⊗Z[G]RpSel_(Sigma)^(Sigma^('))(H)_(p)^(-)=Sel_(Sigma)^(Sigma^('))(H)ox_(Z[G])R_(p)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}=\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H) \otimes_{\mathbf{Z}[G]} R_{p}SelΣΣ′â¡(H)p−=SelΣΣ′â¡(H)⊗Z[G]Rp. We have
Since SelS∞T(H)−≅ClT(H)−,vSelS∞Tâ¡(H)−≅ClT(H)−,vSel_(S_(oo))^(T)(H)^(-)~=Cl^(T)(H)^(-,v)\operatorname{Sel}_{S_{\infty}}^{T}(H)^{-} \cong \mathrm{Cl}^{T}(H)^{-, v}SelS∞Tâ¡(H)−≅ClT(H)−,v, it then remains to calculate the effect of removing the primes v∈Sram ,v∣pv∈Sram ,v∣pv inS_("ram "),v∣pv \in S_{\text {ram }}, v \mid pv∈Sram ,v∣p from ΣΣSigma\SigmaΣ. This is a delicate process using functorial properties of the Ritter-Weiss modules discussed in §6§6§6\S 6§§6, and one obtains (see [17, APPENDIX B]) the desired result
In this section we summarize the proof of Theorem 5.6.
6.1. Ritter-Weiss modules
The Z[G]Z[G]Z[G]\mathbf{Z}[G]Z[G]-module that shows up in our constructions with modular forms is a certain transpose of SelST(H)SelSTâ¡(H)Sel_(S)^(T)(H)\operatorname{Sel}_{S}^{T}(H)SelSTâ¡(H) in the sense of Jannsen [31], denoted ∇ST(H)∇ST(H)grad_(S)^(T)(H)\nabla_{S}^{T}(H)∇ST(H). This module was originally defined by Ritter and Weiss in the foundational paper [40] without the smoothing set TTTTT. We incorporated the smoothing set TTTTT and established some additional properties of ∇ST(H)∇ST(H)grad_(S)^(T)(H)\nabla_{S}^{T}(H)∇ST(H) in [17, APPENDIX A]. To describe these properties, we work over Rp=Zp[G]−Rp=Zp[G]−R_(p)=Z_(p)[G]^(-)R_{p}=\mathbf{Z}_{p}[G]^{-}Rp=Zp[G]−and consider finite disjoint sets Σ,Σ′Σ,Σ′Sigma,Sigma^(')\Sigma, \Sigma^{\prime}Σ,Σ′ satisfying the following:
Note that the pair (S,T)(S,T)(S,T)(S, T)(S,T) from Section 2 and the pair (Σ,Σ′)Σ,Σ′(Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′) considered in Section 5.3 both satisfy these conditions. The module ∇ΣΣ′(H)p−=∇ΣΣ′(H)⊗Z[G]Rp∇ΣΣ′(H)p−=∇ΣΣ′(H)⊗Z[G]Rpgrad_(Sigma)^(Sigma^('))(H)_(p)^(-)=grad_(Sigma)^(Sigma^('))(H)ox_(Z[G])R_(p)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}=\nabla_{\Sigma}^{\Sigma^{\prime}}(H) \otimes_{\mathbf{Z}[G]} R_{p}∇ΣΣ′(H)p−=∇ΣΣ′(H)⊗Z[G]Rp satisfies the following:
There is a short exact sequence of RpRpR_(p)R_{p}Rp-modules
Here ClΣΣ′(H)ClΣΣ′(H)Cl_(Sigma)^(Sigma^('))(H)\mathrm{Cl}_{\Sigma}^{\Sigma^{\prime}}(H)ClΣΣ′(H) denotes the quotient of ClΣ′(H)ClΣ′(H)Cl^(Sigma^('))(H)\mathrm{Cl}^{\Sigma^{\prime}}(H)ClΣ′(H) by the image of the primes in ΣHΣHSigma_(H)\Sigma_{H}ΣH.
Given a RpRpR_(p)R_{p}Rp-module BBBBB, a surjective RpRpR_(p)R_{p}Rp-module homomorphism
(6.2)∇ΣΣ′(H)p−→B(6.2)∇ΣΣ′(H)p−→B{:(6.2)grad_(Sigma)^(Sigma^('))(H)_(p)^(-)rarr B:}\begin{equation*}
\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \rightarrow B \tag{6.2}
\end{equation*}(6.2)∇ΣΣ′(H)p−→B
is equivalent to the data of a cocycle κ∈Z1(GF,B)κ∈Z1GF,Bkappa inZ^(1)(G_(F),B)\kappa \in Z^{1}\left(G_{F}, B\right)κ∈Z1(GF,B) and a collection of elements xv∈Bxv∈Bx_(v)in Bx_{v} \in Bxv∈B for v∈Σv∈Σv in Sigmav \in \Sigmav∈Σ satsifying the following conditions:
The cohomology class [κ]∈H1(GF,B)[κ]∈H1GF,B[kappa]inH^(1)(G_(F),B)[\kappa] \in H^{1}\left(G_{F}, B\right)[κ]∈H1(GF,B) is unramified outside Σ′Σ′Sigma^(')\Sigma^{\prime}Σ′, tamely ramified at Σ′Σ′Sigma^(')\Sigma^{\prime}Σ′, and locally trivial at ΣΣSigma\SigmaΣ.
The xvxvx_(v)x_{v}xv provide local trivializations at Σ:κ(σ)=(σ−1)xvΣ:κ(σ)=(σ−1)xvSigma:kappa(sigma)=(sigma-1)x_(v)\Sigma: \kappa(\sigma)=(\sigma-1) x_{v}Σ:κ(σ)=(σ−1)xv for σ∈Gvσ∈Gvsigma inG_(v)\sigma \in G_{v}σ∈Gv.
The xvxvx_(v)x_{v}xv along with the image of κκkappa\kappaκ generate the module BBBBB over RpRpR_(p)R_{p}Rp.
The tuples (κ,{xv})κ,xv(kappa,{x_(v)})\left(\kappa,\left\{x_{v}\right\}\right)(κ,{xv}) are taken modulo the natural notion of coboundary, i.e., (κ,{xv})∼(κ+dx,{xv+x})κ,xv∼κ+dx,xv+x(kappa,{x_(v)})∼(kappa+dx,{x_(v)+x})\left(\kappa,\left\{x_{v}\right\}\right) \sim\left(\kappa+d x,\left\{x_{v}+x\right\}\right)(κ,{xv})∼(κ+dx,{xv+x}) for x∈Bx∈Bx in Bx \in Bx∈B.
The module ∇ΣΣ′(H)p−∇ΣΣ′(H)p−grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−has a quadratic presentation, i.e., there exists an exact sequence of RpRpR_(p)R_{p}Rp-modules
where M1M1M_(1)M_{1}M1 and M2M2M_(2)M_{2}M2 are free of the same finite rank.
The module ∇ΣΣ′(H)p−∇ΣΣ′(H)p−grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−is a transpose of SelΣΣ′(H)p−SelΣΣ′â¡(H)p−Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}SelΣΣ′â¡(H)p−, i.e., for a suitable quadratic presentation (6.3) of ∇ΣΣ′(H)p−∇ΣΣ′(H)p−grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−, the cokernel of the induced map
is isomorphic to SelΣΣ′(H)p−SelΣΣ′â¡(H)p−Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}SelΣΣ′â¡(H)p−. Here we follow our convention of giving Hom spaces the contragradient GGGGG-action.
The quadratic presentation property (6.3) implies that FittRp(∇ΣΣ′(H)p−)=det(A)FittRpâ¡âˆ‡Î£Î£â€²(H)p−=detâ¡(A)Fitt_(R_(p))(grad_(Sigma)^(Sigma^('))(H)_(p)^(-))=det(A)\operatorname{Fitt}_{R_{p}}\left(\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\operatorname{det}(A)FittRpâ¡(∇ΣΣ′(H)p−)=detâ¡(A) is principal. The transpose property (6.4) implies that
We now fix (Σ,Σ′)Σ,Σ′(Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′) to be the pair defined in Section 5.3. In the remainder of this section we outline how (6.6) is proved using Ribet's method. Throughout, an unadorned ΘΘTheta\ThetaΘ denotes ΘΣ,Σ′(ΘΣ,Σ′Theta_(Sigma,Sigma^('))(:}\Theta_{\Sigma, \Sigma^{\prime}}\left(\right.ΘΣ,Σ′( and Θ#Θ#Theta^(#)\Theta^{\#}Θ# denotes ΘΣ,Σ′#ΘΣ,Σ′#Theta_(Sigma,Sigma^('))^(#)\Theta_{\Sigma, \Sigma^{\prime}}^{\#}ΘΣ,Σ′#.
6.2. Inclusion implies equality
An interesting feature of Ribet's method is that it tends to prove an inclusion in one direction, that of an algebraically defined ideal contained within an analytically defined ideal. In our setting, we use it to prove
We then employ an analytic argument to show that this inclusion is an equality. It is important to note that the inclusion (6.7) is the reverse direction of that required by the Brumer-Stark and Strong Brumer-Stark conjectures. It is therefore essential for our approach that one actually has the statement of an equality rather than just an inclusion (and an analytic argument to deduce the equality from the reverse inclusion). For this reason, the conjecture of Burns stated in Section 5.3 (more precisely the analog of it stated in Theorem 5.6) plays an essential role in our strategy.
Let us describe the analytic argument in the special case Σ=S∞Σ=S∞Sigma=S_(oo)\Sigma=S_{\infty}Σ=S∞, i.e. there are no primes above ppppp ramified in H/FH/FH//FH / FH/F. In this case
is finite and ΘΘTheta\ThetaΘ is a non-zero-divisor. Using (6.7), write
(6.9)FittRp(SelΣΣ′(H)p−)=(Θ#⋅z) for some z∈Rp(6.9)FittRpâ¡SelΣΣ′â¡(H)p−=Θ#â‹…z for some z∈Rp{:(6.9)Fitt_(R_(p))(Sel_(Sigma)^(Sigma^('))(H)_(p)^(-))=(Theta^(#)*z)quad" for some "z inR_(p):}\begin{equation*}
\operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\left(\Theta^{\#} \cdot z\right) \quad \text { for some } z \in R_{p} \tag{6.9}
\end{equation*}(6.9)FittRpâ¡(SelΣΣ′â¡(H)p−)=(Θ#â‹…z) for some z∈Rp
We must show that z∈Rp∗z∈Rp∗z inR_(p)^(**)z \in R_{p}^{*}z∈Rp∗. An elementary argument (see [17, $2.3]) shows that (6.9) implies
where the subscript ppppp on the right denotes the ppppp-power part of an integer. Yet the analytic class number formula implies (see [17,$2.1][17,$2.1][17,$2.1][17, \$ 2.1][17,$2.1] )
where ≐â‰â‰\doteq≠denotes equality up to a power of 2 . Combining (6.8), (6.10), and (6.11), one finds that χ(z)χ(z)chi(z)\chi(z)χ(z) is a ppppp-adic unit for each odd character χχchi\chiχ. It follows that z∈Rp∗z∈Rp∗z inR_(p)^(**)z \in R_{p}^{*}z∈Rp∗ as desired.
The generalization of this argument to arbitrary ΣΣSigma\SigmaΣ requires a delicate induction and is described in [17,$5][17,$5][17,$5][17, \$ 5][17,$5].
6.3. Ribet's method
We now describe our implementation of Ribet's method to prove the inclusion (6.7). The idea is to use the Galois representations associated to Hilbert modular forms to construct an RpRpR_(p)R_{p}Rp-module MMMMM and a surjection ∇ΣΣ′(H)p−→M∇ΣΣ′(H)p−→Mgrad_(Sigma)^(Sigma^('))(H)_(p)^(-)rarr M\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \rightarrow M∇ΣΣ′(H)p−→M such that FittRp(M)⊂(Θ)FittRpâ¡(M)⊂(Θ)Fitt_(R_(p))(M)sub(Theta)\operatorname{Fitt}_{R_{p}}(M) \subset(\Theta)FittRpâ¡(M)⊂(Θ). The properties of Fitting ideals imply that (6.7) follows from the existence of such a surjection. As described in (6.2), a surjection from ∇ΣΣ′(H)p−∇ΣΣ′(H)p−grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−can be constructed by defining a cohomology class [κ]∈H1(GF,M)[κ]∈H1GF,M[kappa]inH^(1)(G_(F),M)[\kappa] \in H^{1}\left(G_{F}, M\right)[κ]∈H1(GF,M) satisfying certain local conditions along with local trivializations at the places in ΣΣSigma\SigmaΣ.
Ribet's method was described in great detail by Mazur in a beautiful article written for the celebration of Ribet's 60th birthday [36]. We borrow from this the following schematic diagram demonstrating the general path one follows to link LLLLL-values (in our case, the Stickelberger element ΘΘTheta\ThetaΘ ) to class groups (in our case, the Ritter-Weiss module ∇ΣΣ′(H)p−∇ΣΣ′(H)p−grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−).
Let us now trace this path in our application.
6.3.1. LLLLL-functions to Eisenstein series
The key connection between LLLLL-functions and modular forms in Ribet's method is that LLLLL-functions appear as constant terms of Eisenstein series. We now define the space of modular forms in which our Stickelberger element ΘΘTheta\ThetaΘ appears.
for a large value of NNNNN. Let n⊂OFn⊂OFnsubO_(F)\mathfrak{n} \subset O_{F}n⊂OF denote the conductor of H/FH/FH//FH / FH/F. Let Mk(n;Z)Mk(n;Z)M_(k)(n;Z)M_{k}(\mathfrak{n} ; \mathbf{Z})Mk(n;Z) denote the group of Hilbert modular forms for FFFFF of level nnn\mathfrak{n}n with Fourier coefficients in ZZZ\mathbf{Z}Z. For each odd character χχchi\chiχ of GGGGG valued in Cp∗Cp∗C_(p)^(**)\mathbf{C}_{p}^{*}Cp∗, let
denote the subspace of forms of nebentypus χχchi\chiχ. Let
χ:GF→G→Rp∗χ:GF→G→Rp∗chi:G_(F)rarr G rarrR_(p)^(**)\chi: G_{F} \rightarrow G \rightarrow R_{p}^{*}χ:GF→G→Rp∗
denote the canonical character, where the first arrow is projection and the second is induced by G↪Z[G]∗G↪Z[G]∗G↪Z[G]^(**)G \hookrightarrow \mathbf{Z}[G]^{*}G↪Z[G]∗.
Definition 6.1. The space Mk(n,χ;Rp)Mkn,χ;RpM_(k)(n,chi;R_(p))M_{k}\left(\mathfrak{n}, \chi ; R_{p}\right)Mk(n,χ;Rp) of group-ring valued Hilbert modular forms of weight kkkkk and level nnn\mathfrak{n}n over RpRpR_(p)R_{p}Rp consists of those f∈Mk(n;Z)⊗Rpf∈Mk(n;Z)⊗Rpf inM_(k)(n;Z)oxR_(p)f \in M_{k}(\mathfrak{n} ; \mathbf{Z}) \otimes R_{p}f∈Mk(n;Z)⊗Rp such that χ(f)∈Mk(n,χ)χ(f)∈Mk(n,χ)chi(f)inM_(k)(n,chi)\chi(f) \in M_{k}(\mathfrak{n}, \chi)χ(f)∈Mk(n,χ) for each odd character χχchi\chiχ. Let Sk(n,χ;Rp)Skn,χ;RpS_(k)(n,chi;R_(p))S_{k}\left(\mathfrak{n}, \chi ; R_{p}\right)Sk(n,χ;Rp) denote the subspace of cusp forms. We define Mk(n,χ;Frac(Rp))Mkn,χ;Fracâ¡RpM_(k)(n,chi;Frac(R_(p)))M_{k}\left(\mathfrak{n}, \chi ; \operatorname{Frac}\left(R_{p}\right)\right)Mk(n,χ;Fracâ¡(Rp)) and Sk(n,χ;Frac(Rp))Skn,χ;Fracâ¡RpS_(k)(n,chi;Frac(R_(p)))S_{k}\left(\mathfrak{n}, \chi ; \operatorname{Frac}\left(R_{p}\right)\right)Sk(n,χ;Fracâ¡(Rp)) similarly.
Hilbert modular forms fffff are determined by their Fourier coefficients c(m,f)c(m,f)c(m,f)c(\mathfrak{m}, f)c(m,f) indexed by the nonzero ideals m⊂OFm⊂OFmsubO_(F)\mathfrak{m} \subset O_{F}m⊂OF and their constant terms cλ(0,f)cλ(0,f)c_(lambda)(0,f)c_{\lambda}(0, f)cλ(0,f) indexed by λ∈Cl+(F)λ∈Cl+(F)lambda inCl^(+)(F)\lambda \in \mathrm{Cl}^{+}(F)λ∈Cl+(F), the narrow class group of FFFFF. For odd k≥1k≥1k >= 1k \geq 1k≥1, there is an Eisenstein series Ek(χ,1)∈Mk(n,χ)Ek(χ,1)∈Mk(n,χ)E_(k)(chi,1)inM_(k)(n,chi)E_{k}(\chi, 1) \in M_{k}(\mathfrak{n}, \chi)Ek(χ,1)∈Mk(n,χ) whose Fourier coefficients are given by